Statement-1: The number of ways of distributi | vedclass

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(D) For Statement-$1$: The number of ways to distribute $n$ identical items into $r$ distinct boxes such that no box is empty is given by the formula

Vedclass · Accepted Answer

Statement-$1$ is true,Statement-$2$ is true; Statement-$2$ is a correct explanation for Statement-$1$.

Vedclass · Answer

(D) For Statement-$1$: The number of ways to distribute $n$ identical items into $r$ distinct boxes such that no box is empty is given by the formula $^{n-1}C_{r-1}$.Here,$n = 10$ and $r = 4$.So,the number of ways is $^{10-1}C_{4-1} = ^9C_3$.Thus,Statement-$1$ is true.For Statement-$2$: The number of ways to choose $r$ items from $n$ distinct items is given by $^nC_r$.Here,$n = 9$ and $r = 3$,so the number of ways is $^9C_3$.Thus,Statement-$2$ is true.Since the formula used in Statement-$1$ is derived from the concept of choosing gaps (stars and bars method),Statement-$2$ provides the correct explanation for Statement-$1$.

Statement-$1$: The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3$.
Statement-$2$: The number of ways of choosing any $3$ places from $9$ different places is $^9C_3$.

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Statement-$1$: The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3$.Statement-$2$: The number of ways of choosing any $3$ places from $9$ different places is $^9C_3$.