Question

The number of permutations of $n$ dissimilar things taken '$r$' at a time, in which a particular thing always occur is ____

• A. ${}^{(}{}^n{}^{-}{}^1{}^{)}P{}_{(}{}_r{}_{-}{}_1{}_{)}$
• B. (r) ${}^{(}{}^n{}^{-}{}^1{}^{)}P{}_{(}{}_r{}_{-}{}_1{}_{)}$
• C. (r) ${}^{(}{}^n{}^{-}{}^1{}^{)}P{}_r$
• D. (r!) ${}^{(}{}^n{}^{-}{}^1{}^{)}P{}_{(}{}_r{}_{-}{}_1{}_{)}$

Answer 1

The correct option is B (r) ${}^{(}{}^n{}^{-}{}^1{}^{)}P{}_{(}{}_r{}_{-}{}_1{}_{)}$

Here we have "$n$" things and we have to find out the permutation of "$r$" things take at a time such that one particular thing is always coming.

Take an example, you have letters $A$ to $Z$.

Here you have to choose $10$ letters such that $R$ always comes i.e. $R$ is fixed, so we just have $25$ letters and we have to choose $9$ out of them as we have already chosen one $\left(R\right)$.

Similarly, here we have to chose $(r-1)$ from $(n-1)$ which can be done in ${}^{(n-1)}{C}_{(r-1)}$.

Now we have $r$ selection which can be arranged in $r!$ ways

${}^{(n-1)}{C}_{(r-1)}\times r!=\left(r\right)\times {}^{(n-1)}{P}_{(r-1)}$.