The total number of 7-digit numbers that can | vedclass

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(C) Let the digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7 \in \{1, 2, 3\}$ such that $\sum_{i=1}^{7} x_i = 10$.Let $n_1, n_2, n_3$ be the number of tim

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$77$

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(C) Let the digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7 \in \{1, 2, 3\}$ such that $\sum_{i=1}^{7} x_i = 10$.Let $n_1, n_2, n_3$ be the number of times $1, 2,$ and $3$ appear respectively.We have $n_1 + n_2 + n_3 = 7$ and $1n_1 + 2n_2 + 3n_3 = 10$.Subtracting the first from the second: $n_2 + 2n_3 = 3$.Case $1$: If $n_3 = 1$,then $n_2 = 1$ and $n_1 = 5$. The number of permutations is $\frac{7!}{5!1!1!} = 42$.Case $2$: If $n_3 = 0$,then $n_2 = 3$ and $n_1 = 4$. The number of permutations is $\frac{7!}{4!3!0!} = 35$.Total numbers = $42 + 35 = 77$.

The total number of $7$-digit numbers that can be formed using only the digits $1, 2,$ and $3$ such that the sum of the digits is $10$ is:

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