JEE Advanced 2023 Paper 2 · Mathematics · Permutations and Combinations · medium

The number of ordered triples (a,b,c) of integers with 0≤ a,b,c≤7 and a+b+c=15 is

Numerical answer

The number of ordered triples

(a,b,c)

of integers with

0\le a,b,c\le7

and

a+b+c=15

Show answer and solution

Answer: 28

Without upper bounds the count is

\binom{17}{2}=136

. Subtract triples with some variable

\ge8

: put

a'=a-8

, giving

a'+b+c=7

\binom92=36

solutions per variable, times 3

=108

. Two variables

\ge8

would need sum

\ge16>15

, so no overlap. Total

136-108=28

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