The number of seven-digit positive integers f | vedclass

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Question

(C) Let the seven digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ where $x_i \in \{1, 2, 3, 4\}$.We need to find the number of solutions to the equation

Vedclass · Accepted Answer

$413$

Vedclass · Answer

(C) Let the seven digits be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ where $x_i \in \{1, 2, 3, 4\}$.We need to find the number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 12$.Let $y_i = x_i - 1$,where $y_i \in \{0, 1, 2, 3\}$.Substituting $x_i = y_i + 1$,we get $(y_1 + 1) + (y_2 + 1) + \dots + (y_7 + 1) = 12$,which simplifies to $y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 = 5$.The number of non-negative integer solutions to this equation is given by the stars and bars formula: $\binom{n+k-1}{k-1} = \binom{5+7-1}{7-1} = \binom{11}{6} = 462$.However,we must satisfy the constraint $x_i \le 4$,which implies $y_i \le 3$.We use the Principle of Inclusion-Exclusion to subtract cases where at least one $y_i \ge 4$.If one $y_i \ge 4$,let $y_i = z_i + 4$. Then $z_i + 4 + \sum_{j 
eq i} y_j = 5$,so $z_i + \sum_{j 
eq i} y_j = 1$.The number of solutions for a fixed $i$ is $\binom{1+7-1}{7-1} = \binom{7}{6} = 7$.Since there are $7$ choices for $i$,we subtract $7 	imes 7 = 49$.Thus,the total number of valid integers is $462 - 49 = 413$.

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

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