Statement-1: 10 identical balls can be distri | vedclass

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Question

(C) The number of ways to distribute $n$ identical items into $r$ distinct boxes such that no box is empty is given by the formula $^{n-1}C_{r-1}$.Her

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) 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 $^{10-1}C_{4-1} = ^9C_3$.Statement-$1$ is true.Statement-$2$ states that $3$ positions out of $9$ can be selected in $^9C_3$ ways,which is the definition of combinations $(^nC_r)$.This logic is exactly what is used in the stars and bars method to arrive at the formula for Statement-$1$.Therefore,Statement-$2$ is the correct explanation for Statement-$1$.

Statement-$1$: $10$ identical balls can be distributed into $4$ distinct boxes in $^9C_3$ ways such that no box remains empty.
Statement-$2$: Any $3$ positions out of $9$ positions can be selected in $^9C_3$ ways.

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Statement-$1$: $10$ identical balls can be distributed into $4$ distinct boxes in $^9C_3$ ways such that no box remains empty.Statement-$2$: Any $3$ positions out of $9$ positions can be selected in $^9C_3$ ways.