Question

The number of selections of $4$ letters from the letter of word $MISSISSIPPI$ is ____

Answer 1

$\begin{array}{cc}\begin{array}{c}M-1\\ 1-4\\ 3-4\\ P-2\end{array}\}& weneed4letterwords.Thereare5casesmtotal\end{array}$

Case $\left(i\right):$ when One letter is repeated $4$ times $IIII$ in $SSSS$

$3$ possibilities

Case $\left(ii\right):$ when one is repeated $3$ times and other one $I$ or $3$ can be repeated $3$ times.

so we selects $1$ of $I$ and $S$ and selectt one of remaining $3$ letters and arrange.

no. of ways ${}^2{C}_1\times \frac{46}{3!}\Rightarrow 4\times 2\Rightarrow 8$ ways

Case $\left(iii\right):$ when one letters is repeated twice and another letter is repeated twice $I,S$ and $P$ can be repeated twice

no. of ways:

${}^3{C}_2\times \frac{4!}{2!2!}\Rightarrow 3\times 6\Rightarrow 18$ ways.

Case $\left(iv\right):$ when one letter is repeated twice and other $2$ places can $2$ difference letters.

$I,S,P$ can be repeated twice.

no. of ways: $^3C_1 \times ^3C_2\times \dfrac {4!}{2!}\ \Rightarrow \ 3\times

\times 12\Rightarrow \ 108$

Case $\left(v\right):$ all letters can distance ; $4$ difference letters $I,S,P,M$

$4!$ ways $=24$

$\therefore$ Total no. of $4$ letter words $=24+108+18+8+2$

$=176$