Combinations Aur Powers | Class 8 Maths Book 1 Ch 2

📘 Class 8 · Ganita Prakash Book 1Ch 2 · Power PlayPost 4 / 9⏱ 25–30 min

Combinations Aur Powers 🔐

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

3 × 4 = ?
10³ = ?
26 × 26 = ?

Answers dekho

12
1,000 (10 × 10 × 10)
676 (26 × 26 = 26 × 20 + 26 × 6 = 520 + 156)

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 2 — Exponent Aur Base Kya Hain? mein powers dobara dekh lijiye.

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🎬 Kahani: Cycle Ka Lock Bhool Gaye!

Golu ki nayi cycle ka 3 pahiye wala lock tha (har pahiye pe 0–9). Usne code set kiya aur… bhool gaya! 😩

Meenu: “Sab try kar lo. 000 se 999 tak.” Golu: “Kitne hain? Lakhon? Poori chhuttiyan chali jaayengi!”

Meenu ne kaha: “Ruko. Pehle aasaan version socho. Agar lock mein sirf 1 pahiya hota?”

Golu: “10 option.” Meenu: “2 pahiye?” Golu soch mein pad gaya… Post ke end tak hum jaanenge ki Golu ko kitni der lagegi, aur agla lock kaisa lena chahiye. ⏳

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📖 Concept: Choices Ka Guna

💡 Is post ki ek badi baat: Jab ek ke baad ek choices karni hon (“yeh aur woh”), to har nayi choice pichle saare combinations ko utne guna kar deti hai. Har jagah options same hon to yeh guna ek power ban jaata hai: options^jagah.

Step 1: Do cheezon ke jode

Meenu ke paas 3 kurte aur 4 dupatte.

Tree diagram: 3 kurte, har ek ke saath 4 dupatte (D1 se D4), kul 12 outfit

Kurta 1 ke saath 4 dupatte, Kurta 2 ke saath 4, Kurta 3 ke saath 4: 4 + 4 + 4 = 3 × 4 = 12 outfit.

WHY guna? Har pehli choice ke liye doosri choice ki poori list repeat hoti hai. Same list baar-baar jodna = guna. Ulta socho to bhi: har dupatte ke saath 3 kurte → 4 × 3 = 12. Same!

Step 2: Teen ya zyada cheezein

3 kurte, 4 dupatte, 2 jodi chappal: pehle 12 outfit, har outfit ke saath 2 chappal → 12 × 2 = 24.

$\text{Kul tareeke} = (\text{choice 1}) \times (\text{choice 2}) \times (\text{choice 3}) \times \dots$

WHY? Tree mein har nayi choice har purani shaakha ko utni nayi shaakhaon mein baant deti hai. Shaakhayein guna hoti jaati hain.

Step 3: Jab har jagah same choices: power!

Lock ke har pahiye pe 10 digit:

Pahiye	Codes	Kaise
1	10	0–9
2	100	har pehle digit ke saath 10 → 10 × 10
3	1,000	har 2-digit code ke saath 10 → $10^3$
4	10,000	$10^4$ (ATM PIN)

$\text{Kul codes} = (\text{har jagah ke options})^{\text{jagah}}$

WHY power? Same number (10) baar-baar guna = power. Base = options, exponent = jagah. Aur check: 3 pahiye = 000 se 999 = theek 1,000 numbers ✅

Step 4: Teen tareeke: kab kaunsa?

Tareeka A (list/tree): sab likh do. Sirf jab options bahut kam hon (≤ 15), ya check karna ho.

Tareeka B (guna ka niyam): har jagah ke options guna karo. Zyaadatar sawaalon ke liye.

Tareeka C (ulta gino): “kam se kam ek 7 wale codes?” Seedha ginna mushkil. Kul codes − bina 7 wale codes = 1,000 − $9^3$ = 1,000 − 729 = 271.

WHY C kaam karta hai? Har code ya to 7 rakhta hai ya nahi rakhta; dono milke kul ban jaate hain. “Kam se kam ek” dikhe to C yaad karo!

Step 5: Simpler case wali trick

Meenu ki trick: mushkil sawaal dikhe to chhota version pehle (1 pahiya, 2 pahiye…), pattern dekho, phir badha do.

WHY kaam karti hai? Chhote case ko list karke check kar sakte ho, to pattern pe bharosa hota hai. Pattern (har baar × 10) samajh aate hi bada case formula se. Yeh trick poori maths mein kaam aati hai!

🪜 Atak Gaye? Aise Socho

Har jagah/kadam ka ek dabba banao aur usme likho kitne options. Phir sab dabbe guna.
“Aur” (dono chahiye) → guna. “Ya” (ek hi chunna) → jod.
Lock/PIN mein 0 bhi gino: 0–9 = 10 options.
Koi pabandi ho (pehla digit 0 nahi) to us dabbe mein options kam karo.
Bada sawaal? 1 aur 2 jagah wala version list karke pattern dekho.

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🔐 Deep Concept: Lambai Badhaao Ya Options? Exponent Hi Raja Hai

Do tareeke se lock “zyada safe” bana sakte ho: har pahiye pe zyada options (base), ya zyada pahiye (exponent). Kaunsa behtar?

Lock	Codes
4 pahiye, digits (10)	$10^4 = 10{,}000$
4 pahiye, letters (26)	$26^4 ≈ 4.6$ lakh
8 pahiye, digits (10)	$10^8$ = 10 crore

Options 10 → 26 kiye (2.6 guna) to codes ≈ 46 guna. Par pahiye 4 → 8 kiye to codes 10,000 guna!

WHY? Base badhaane se har jagah thoda zyada milta hai. Exponent badhaane se poora guna ek baar aur lagta hai, aur woh har baar 10,000 jaise bade number ko phir guna karta hai.

Ek computer 1 second mein 1,000 password try kare to: 4 digit PIN = 10 second; 8 character password (A–Z, a–z, 0–9 = 62 options) = $62^8 ≈ 2.2 \times 10^{14}$ → lagbhag 7,000 saal! 🔒

Unifying idea: combinations bhi exponential growth hi hain (Post 1 — Exponential Growth Kya Hai?): har nayi jagah = ek aur “× options”. Isliye websites kehti hain: “password lamba rakho”.

sirf charumam.com readers ke liye 🔐

✏️ Solved Examples

Example 1 (Easy): 4 shirt aur 5 pant. Kitne alag outfit?

Do dabbe: [4] × [5] = 20.

Check: shirt 1 ke saath 5, aise 4 shirt → 5 + 5 + 5 + 5 = 20 ✅

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Example 2 (Medium): Ek suitcase lock mein 4 pahiye, har pe 0–9. Kitne codes? Kitne mein kam se kam ek 0 hai?

Kul = 10⁴ = 10,000.

Bina 0 ke: har pahiye pe 9 options → 9⁴ = 6,561. Kam se kam ek 0: 10,000 − 6,561 = 3,439.

Check: 6,561 + 3,439 = 10,000 ✅ (har code kisi ek group mein)

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Example 3 (NCERT style): School ID code: pehle 2 capital letters (A–Z), phir 3 digit. Kitne ID?

Dabbe: [26][26][10][10][10] = 26² × 10³ = 676 × 1,000 = 6,76,000.

Check: letters ke 676 jode, har jode ke saath 000–999 = 1,000 number → 676 × 1,000 ✅

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Golu ka hisaab: 1,000 codes, ek code 3 second → 3,000 s = 50 minute (sabse buri kismat mein). Golu ne lock khola aur bola: “Agli baar 4 pahiye wala, aur code diary mein!” 🚲

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🏠 Real Life Mein Combinations

🏧 ATM PIN: 4 digit = 10⁴ code. Isliye 3 galat try ke baad card block hota hai.
📮 PIN code: 6 digit, pehla digit 1–9 (0 nahi) → 9 × 10⁵ = 9 lakh possible codes.
🚗 Gaadi ka number: jaise MH 12 AB 1234: letters aur digits ke dabbe guna → har district mein lakhon number.
🍱 Thaali menu: 3 sabzi × 2 daal × 4 roti/chawal = 24 alag thaali. Roz nayi!
🧬 DNA: 4 “letters” (A, T, G, C); sirf 10 letter lambi line ke 4¹⁰ ≈ 10 lakh roop. Isliye har jeev alag!

Sawaal ho “kitne tareeke”, to dabbe banao, options likho, guna karo.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Har kadam alag, ek ke baad ek choice	✅ Options guna karo	Tree ki shaakhayein
Har jagah same options (lock, PIN)	✅ Power: options^jagah	Same guna baar-baar
Ya-to-ya (shirt YA kurta)	❌ Guna nahi, jodo	Ek hi choice hoti hai
“Kam se kam ek …”	✅ Kul − “ek bhi nahi”	Ulta ginna aasaan
Options bahut kam, check karna hai	✅ List/tree	Har case dikhta hai

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❌→✅ Worked Mistake

Sawaal: 3 pahiye ka lock (0–9 har pahiye). Kitne codes?

❌ Galat 1: “10 + 10 + 10 = 30.” ❌ Galat 2: “3¹⁰.”

Galti kya hai? Pehle mein guna ki jagah jod: par har pehle digit ke saath poore 10 doosre digit lagte hain. Doosre mein base-exponent ulat diye: 3 jagah, 10 option → 10³.

✅ Sahi: 10 × 10 × 10 = 1,000. Check: codes 000 se 999 tak = theek 1,000 numbers ✅

Doosri common galti: 3 digit numbers (100–999) ko 1,000 bolna. Number ka pehla digit 0 nahi ho sakta: 9 × 10 × 10 = 900.

🧠 Rule of thumb: Base = har jagah ke options, exponent = kitni jagah. “Options ki power jagah.”

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📝 10 MCQs

Q1. 5 topi, 2 chashme. Kitne jode?

(a) 7

(b) 10

(c) 25

(d) 32

Solution dekho

Sahi jawab: (b)

(a) galat: 5 + 2 jod diya; har topi ke saath dono chashme lagte hain.
(b) sahi: Har topi ke saath 2 → 5 × 2 = 10 ✅
(c) galat: 5² kar diya; chashme 2 hain, 5 nahi.
(d) galat: 2⁵ kar diya; yahan sirf do dabbe hain.

Q2. 2 pahiye lock (0–9). Kitne codes?

(a) 20

(b) 99

(c) 100

(d) 19

Solution dekho

Sahi jawab: (c)

(a) galat: 10 + 10 jod diya; guna chahiye.
(b) galat: 00 bhool gaye: 00 se 99 tak 100 hain.
(c) sahi: 10 × 10 = 100 ✅ (00 se 99)
(d) galat: Koi niyam nahi, andaaza galat.

Q3. 4 digit PIN mein kitne options?

(a) 40

(b) 4¹⁰

(c) 10⁴

(d) 9,999

Solution dekho

Sahi jawab: (c)

(a) galat: 10 × 4 jod jaisa liya; har jagah guna hoti hai.
(b) galat: Base aur exponent ulat diye.
(c) sahi: 4 jagah, har ek pe 10 → 10⁴ ✅
(d) galat: 0000 bhi ek PIN hai, to kul 10,000.

Q4. 3 letter code (A–Z):

(a) 78

(b) 26³

(c) 3²⁶

(d) 26 × 3

Solution dekho

Sahi jawab: (b)

(a) galat: 26 × 3 = 78, jod jaisa soch.
(b) sahi: 3 jagah, har pe 26 → 26³ ✅
(c) galat: Base aur exponent ulte kar diye.
(d) galat: 3 se guna kiya, power nahi lagayi.

Q5. Thaali: 3 sabzi, 2 daal, 2 meetha. Kitni thaali?

(a) 7

(b) 12

(c) 6

(d) 18

Solution dekho

Sahi jawab: (b)

(a) galat: Jod diya; teeno cheezein ek saath aati hain.
(b) sahi: Teen dabbe: 3 × 2 × 2 = 12 ✅
(c) galat: Ek dabba (meetha) chhoot gaya.
(d) galat: 3 × 2 × 3: meetha ke options galat liye.

Q6. 4 pahiye → 5 pahiye lock: codes kitne guna?

(a) 2 guna

(b) 5 guna

(c) 10 guna

(d) 100 guna

Solution dekho

Sahi jawab: (c)

(a) galat: Doubling nahi; har pahiye pe 10 options hain.
(b) galat: 5 pahiye hain, par guna options se hota hai.
(c) sahi: Ek pahiya = ek aur × 10 ✅
(d) galat: 100 guna tab hota jab do pahiye badhte.

Q7. 3 digit numbers (100–999) kitne?

(a) 1,000

(b) 900

(c) 999

(d) 899

Solution dekho

Sahi jawab: (b)

(a) galat: 000–099 bhi gin liye, jo 3 digit number nahi.
(b) sahi: Pehla digit 1–9 (9 options) → 9 × 10 × 10 = 900 ✅
(c) galat: 999 − 100 = 899 karke 1 joda, phir bhi galat.
(d) galat: Ek kam gina: 100 aur 999 dono shaamil hain.

Q8. Kaun sa “ya” wala sawaal hai (jod lagega)?

(a) Shirt AUR pant chunna

(b) Kurta YA saree (ek hi pehnna)

(c) Lock ke 3 pahiye

(d) PIN ke 4 digit

Solution dekho

Sahi jawab: (b)

(a) galat: Dono pehnte ho, to guna lagega.
(b) sahi: Sirf ek choice: kurte ke options + saree ke options ✅
(c) galat: Har pahiya alag choice, guna.
(d) galat: Har digit alag choice, guna.

Q9. Code mein 2 jagah, har jagah 36 option (A–Z, 0–9). Kul?

(a) 72

(b) 1,296

(c) 36

(d) 38

Solution dekho

Sahi jawab: (b)

(a) galat: 36 + 36 jod diya; guna chahiye.
(b) sahi: Do jagah, har pe 36: 36² = 1,296 ✅
(c) galat: Sirf ek jagah gini, doosri bhool gaye.
(d) galat: 36 + 2: jagah ko option maan liya.

Q10. 3 digit lock: kitne codes mein kam se kam ek 5 hai?

(a) 300

(b) 271

(c) 729

(d) 100

Solution dekho

Sahi jawab: (b)

(a) galat: 3 × 100: har jagah ke 5 ko alag gina, repeat double gine.
(b) sahi: Kul − bina 5 = 1,000 − 9³ = 1,000 − 729 = 271 ✅
(c) galat: 729 to bina 5 wale codes hain, ulta jawab.
(d) galat: Sirf pehli jagah 5 wale gine (500–599).

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Jod kab, guna kab?

“Aur” (dono/sab chahiye) → guna. “Ya” (sirf ek) → jod. 3 kurte ya 2 saree pehnne ke 5 tareeke; 3 kurte aur 2 dupatte ke 6.

2. 000 bhi code hai?

Haan! Lock pe 000 bhi ek setting hai. Isliye 3 pahiye = 1,000. Par 3 digit number mein 0 aage nahi aata, isliye 900.

3. PIN code ka pehla digit 0 kyun nahi?

India mein pehla digit zone batata hai (1–8, aur 9 Army Post Office ke liye). 0 kisi zone ka nahi, isliye us dabbe mein 9 options.

4. Kya order matter karta hai?

Lock/PIN mein haan (123 ≠ 321). Kuch sawaalon mein nahi (jaise team mein kaun-kaun); unka hisaab alag hai, badi class mein padhoge.

5. Tree diagram kab banayein?

Jab options kam hon aur aap samajhna ya check karna chahte ho. Bade case mein seedha guna.

6. 3 dost 3 kursi pe: 3³ kyun nahi?

Kyunki ek dost ek baar baith gaya to agli kursi ke liye options ghat jaate hain: 3 × 2 × 1 = 6. Power tabhi jab har jagah options same rahein (lock mein digit repeat ho sakta hai).

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy 6 shirt, 3 pant. Kitne outfit?

Solution dekho

6 × 3 = 18. Check: har pant ke saath 6 shirt → 6 + 6 + 6 ✅

Q2. Easy 5 pahiye lock (0–9). Codes?

Solution dekho

10⁵ = 1,00,000. Check: 00000 se 99999 = 1 lakh ✅

Q3. Easy 2 letter code (A–Z)?

Solution dekho

26² = 676. Check: AA se AZ = 26, aise 26 pehle letter ✅

Q4. Easy Ek dice 2 baar phenka. Kitne result?

Solution dekho

6 × 6 = 36. Check: (1,1) se (6,6) tak ki 6 × 6 table ✅

Q5. Medium Ek coin 5 baar uchhala (H ya T). Kitne pattern?

Solution dekho

2⁵ = 32. Check: 2 baar → HH, HT, TH, TT = 4 = 2² ✅ pattern sahi

Q6. Medium 2 digit numbers (10–99) kitne?

Solution dekho

9 × 10 = 90. Check: 99 − 10 + 1 = 90 ✅

Q7. Medium Ice cream: 4 flavour, 3 cone, topping ho ya na ho. Kitne?

Solution dekho

4 × 3 × 2 = 24. WHY 2: topping ka dabba = haan/nahi.

Q8. Medium Code: 1 letter (A–Z) + 2 digit. Kitne?

Solution dekho

26 × 10² = 2,600. Check: A00–A99 = 100, aise 26 letter ✅

Q9. NCERT style Ek tijori 4 digit ke code se khulti hai. Har 2 second mein ek code try karo to sab try karne mein max kitna samay?

Solution dekho

10,000 × 2 = 20,000 s ≈ 5.6 ghante. Check: 20,000 ÷ 3,600 ≈ 5.56 ✅

Q10. NCERT style Mobile number 10 digit, pehla digit 6, 7, 8 ya 9. Kitne number possible?

Solution dekho

4 × 10⁹ = 400 crore. WHY: pehle dabbe mein 4, baaki 9 dabbon mein 10-10.

Q11. NCERT style Ek school mein roll code: section (A–E) + roll no. 01–60. Kitne code?

Solution dekho

5 × 60 = 300. WHY power nahi: dono dabbon ke options alag (5 aur 60).

Q12. NCERT style Binary code: 8 jagah, har pe 0 ya 1. Kitne codes?

Solution dekho

2⁸ = 256. Isliye 1 byte = 256 values! Check: 00000000 se 11111111 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Lock A: 3 pahiye, har pe 10 digit. Lock B: 2 pahiye, har pe 26 letter. Kaunsa zyada safe?

Solution dekho

A: 10³ = 1,000. B: 26² = 676. A. WHY: exponent (jagah) ka asar base se zyada.

Q2. Kitne pahiye (0–9) chahiye ki 50 lakh se zyada codes bane?

Solution dekho

10⁶ = 10 lakh (kam), 10⁷ = 1 crore ✅ → 7 pahiye.

Q3. Ek dairy saal mein 30 crore packet pe alag ID chahti hai, sirf digits (0–9) se. Kam se kam kitne digit?

Solution dekho

10⁸ = 10 crore (kam), 10⁹ = 100 crore ✅ → 9 digit.

Q4. 3 dost 3 kursiyon pe kitne tareeke se baith sakte hain? List karke check karo.

Solution dekho

3 × 2 × 1 = 6: ABC, ACB, BAC, BCA, CAB, CBA ✅. Har kadam pe options kam hote gaye, isliye power nahi.

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🏆 Olympiad Corner

Q1. 2 digit codes (00–99) mein kitne mein dono digit alag hain?

Solution dekho

Pehla 10, doosra (pehle wala chhod ke) 9 → 90. Check: same wale 00, 11, …, 99 = 10; 100 − 10 = 90 ✅

Q2. 4 bulb, har ek ON ya OFF. Kitne pattern mein kam se kam ek ON?

Solution dekho

2⁴ − 1 (sab OFF) = 15.

Q3. Kitne 4 digit numbers sirf 1, 2, 3 se ban sakte hain (repeat allowed)?

Solution dekho

3⁴ = 81.

Q4. 1 se 999 tak kitne numbers mein digit 7 nahi aata?

Solution dekho

000–999 mein bina 7: 9³ = 729 (isme 000 bhi). 000 hatao → 728.

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💗 CharuMam Motivation

Golu ghabra gaya tha: “lakhon codes!” Par Meenu ne chhote se shuru kiya aur 1 minute mein pata chal gaya: sirf 1,000. Bade dar aksar chhote sawaalon mein toot jaate hain. 🪜

Aur yaad rakho: aapke dimaag mein ideas ke combinations ki koi limit nahi. Har nayi cheez jo seekhte ho, purani cheezon se judke naye raaste kholti hai. Seekhte raho! 🌟

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🟢 Master Recap

💡 Chaabi: Jab ek ke baad ek choices karni hon (“yeh aur woh”), to har nayi choice pichle saare combinations ko utne guna kar deti hai. Har jagah options same hon to yeh guna ek power ban jaata hai: options^jagah.
“Aur” wali choices → guna; “ya” wali → jod.
3 kurte × 4 dupatte = 12 outfit (tree se check).
Har jagah same options → options^jagah.
3 pahiye = 10³ = 1,000; 4 digit PIN = 10⁴.
“Kam se kam ek” → kul − “ek bhi nahi”.
Exponent (lambai) badhaana base badhaane se zyada asar karta hai.
Mushkil sawaal → chhota version pehle.

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👉 Aage Kya?

Ab tak powers sirf guna se badh rahi thi. Par agar hum bhaag karein to? 2⁰ kya hoga, aur 2⁻³ ka matlab? Agle post mein: zero aur negative exponent! ➖

← Pichla: Post 3 — Exponent Ke Niyam: Guna Aur Power 📚 Chapter 2 ke saare posts Agla: Post 5 — Zero Aur Negative Exponent →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Cycle Ka Lock Bhool Gaye!

📖 Concept: Choices Ka Guna

Step 1: Do cheezon ke jode

Step 2: Teen ya zyada cheezein

Step 3: Jab har jagah same choices: power!

Step 4: Teen tareeke: kab kaunsa?

Step 5: Simpler case wali trick

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Lambai Badhaao Ya Options? Exponent Hi Raja Hai

✏️ Solved Examples

🏠 Real Life Mein Combinations

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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