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Permutations and Combinations question

2025 · 28 Jan · Shift 2 · Q50
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Permutations and Combinations question

2025 · 28 Jan · Shift 2 · Q50

JEE MainMathematicsPermutations and CombinationsNumerical+4 / −1
The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 64

  1. Let the three-digit number be written as abcabcabc, where:

    • aaa is the hundreds digit,
    • bbb is the tens digit,
    • ccc is the units digit.

    We need: a+b+c=15a+b+c=15a+b+c=15

  2. Also, the number must lie between 212212212 and 999999999.

    Since every number from 212212212 to 999999999 is a three-digit number, we count valid digit triples (a,b,c)(a,b,c)(a,b,c) with: 1≤a≤9,0≤b,c≤91\le a\le 9,\quad 0\le b,c\le 91≤a≤9,0≤b,c≤9 and then exclude those less than 212212212.

  3. First count all three-digit numbers whose digits sum to 151515.

    We solve: a+b+c=15a+b+c=15a+b+c=15 with a≥1a\ge 1a≥1, b,c≥0b,c\ge 0b,c≥0.

    Put: a′=a−1≥0a'=a-1\ge 0a′=a−1≥0 Then: a′+b+c=14a'+b+c=14a′+b+c=14

    Number of non-negative integer solutions is: (14+3−13−1)=(162)=120\binom{14+3-1}{3-1}=\binom{16}{2}=120(3−114+3−1​)=(216​)=120

  4. Now check digit upper bounds a,b,c≤9a,b,c\le 9a,b,c≤9.

    Since a+b+c=15a+b+c=15a+b+c=15, at most one variable can exceed 999.

    • If a≥10a\ge 10a≥10, let a′′=a−10≥0a''=a-10\ge 0a′′=a−10≥0. Then: a′′+b+c=5a''+b+c=5a′′+b+c=5 Number of solutions: (5+3−12)=(72)=21\binom{5+3-1}{2}=\binom{7}{2}=21(25+3−1​)=(27​)=21

    • If b≥10b\ge 10b≥10, let b′′=b−10≥0b''=b-10\ge 0b′′=b−10≥0. Then: a+b′′+c=5a+b''+c=5a+b′′+c=5 with a≥1a\ge 1a≥1. Put a′=a−1≥0a'=a-1\ge 0a′=a−1≥0: a′+b′′+c=4a'+b''+c=4a′+b′′+c=4 Number of solutions: (4+3−12)=(62)=15\binom{4+3-1}{2}=\binom{6}{2}=15(24+3−1​)=(26​)=15

    • If c≥10c\ge 10c≥10, similarly number of solutions is: 151515

    Hence valid three-digit numbers with digit sum 151515 are: 120−21−15−15=69120-21-15-15=69120−21−15−15=69

  5. Now exclude those numbers less than 212212212.

    Such numbers are from 100100100 to 211211211. We need digit sum 151515.

    • If a=1a=1a=1, then: b+c=14b+c=14b+c=14 Possible digit pairs are: (5,9),(6,8),(7,7),(8,6),(9,5)(5,9),(6,8),(7,7),(8,6),(9,5)(5,9),(6,8),(7,7),(8,6),(9,5) giving numbers: 159,168,177,186,195159,168,177,186,195159,168,177,186,195 These are all <212<212<212.

    • If a=2a=2a=2, then for numbers <212<212<212, we only have 200200200 to 211211211. Their digit sums are at most 2+1+1=42+1+1=42+1+1=4, so none has digit sum 151515.

    Therefore, exactly 555 such numbers are below 212212212.

  6. Required count: 69−5=6469-5=6469−5=64

Therefore, the number of natural numbers between 212212212 and 999999999 whose digits sum to 151515 is: 64\boxed{64}64​

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