The number of seven-digit integers with the s | vedclass

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Question

(C) Let the seven digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ such that $x_i \in \{1, 2, 3\}$ and $\sum_{i=1}^{7} x_i = 10$.Case $I$: The digits are

Vedclass · Accepted Answer

$77$

Vedclass · Answer

(C) Let the seven digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ such that $x_i \in \{1, 2, 3\}$ and $\sum_{i=1}^{7} x_i = 10$.Case $I$: The digits are $1, 1, 1, 1, 1, 2, 3$.Sum $= 1+1+1+1+1+2+3 = 10$.The number of arrangements is $\frac{7!}{5!1!1!} = \frac{7 	imes 6}{1} = 42$.Case $II$: The digits are $1, 1, 1, 1, 2, 2, 2$.Sum $= 1+1+1+1+2+2+2 = 10$.The number of arrangements is $\frac{7!}{4!3!} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = 35$.Total number of such integers $= 42 + 35 = 77$.

The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only is:

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