The number of permutations of n things taken | vedclass

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(C) To find the number of permutations of $n$ things taken $r$ at a time such that $p$ specific things are always included:$1$. First,we select the $r

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$^{n-p}C_{r-p} \cdot r!$

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(C) To find the number of permutations of $n$ things taken $r$ at a time such that $p$ specific things are always included:$1$. First,we select the $r$ objects. Since $p$ objects must always be included,we need to choose the remaining $r-p$ objects from the remaining $n-p$ objects. This can be done in $^{n-p}C_{r-p}$ ways.$2$. Once the $r$ objects are selected,they can be arranged among themselves in $r!$ ways.$3$. Therefore,the total number of permutations is $^{n-p}C_{r-p} 	imes r!$.

The number of permutations of $n$ things taken $r$ at a time,when $p$ things are always included,is

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