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JEE 2023 Mathematics PYQ — The number of seven digit positive integers formed using the digi… | Mathem Solvex | Mathem Solvex

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The number of seven digit positive integers formed using the digits $1, 2, 3$ and $4$ only and sum of the digits equal to $12$ is ________.

Choose the correct answer:

A.
413
(Correct Answer)
B.
412
C.
411
D.
410

Correct Answer:

413

Explanation

x_1 + x_2 + \dots + x_7 = 12, \quad x_i \in \{1, 2, 3, 4\}

\text{Hence, no. of required solutions are}

= {}^{12-1}C_{7-1} - \frac{7!}{6!} - \frac{7!}{5!}

= {}^{11}C_{6} - 7 - 7 \times 6

= \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2} - 7 - 42 = 462 - 49 = 413

Explanation

x_1 + x_2 + \dots + x_7 = 12, \quad x_i \in \{1, 2, 3, 4\}

\text{Hence, no. of required solutions are}

= {}^{12-1}C_{7-1} - \frac{7!}{6!} - \frac{7!}{5!}

= {}^{11}C_{6} - 7 - 7 \times 6

= \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2} - 7 - 42 = 462 - 49 = 413

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