Question

Number of words that can be formed by using the $4$ letters of the word $MISSISSIPPI$ is

• A. $192$
• B. $148$
• C. $176$
• D. $164$

Answer 1

The correct option is C $176$

We are given the word $MISSISSIPPI$
Here,
$M-1$
$I-4$
$S-4$
$P-2$

Case $1:$ When all the four letters are distinct:-
${}^4{P}_4=4!=24$

Case $2$: When two letters are repeated:-
the repeated letter can be selected from $I,S,P$ i.e. ${}^3{C}_1$ and the remaining two letters can be selected from $3$ i.e. ${}^3{C}_2.$ After selecting these $4$ letters we have to arrange them while keep in mind that $2$ of them are same.
Number of ways
$=({}^3{C}_1\times {}^3{C}_2)\frac{4!}{2!}$
$=108$

Case $3$: When one is repeated twice and another one is repeated twice:-
Numbe rof ways
$={}^3{C}_2\times \frac{4!}{2!2!}$
$=18$

Case $4$: When one is repeated thrice and other is once:-
Number of ways:
$=({}^2{C}_1\times {}^3{C}_1)\frac{4!}{3!}$
$=2\times 3\times 4$
$=24$

Case $5$: When one is repeated $4$ times:-
Number of ways:
$={}^2{C}_1\times \frac{4!}{4!}$
$=2$

Total number of $4$ letter words formed from the letters of the words $MISSISSIPPI$ can be computed by summing up the result of all these $5$ cases.

i.e. $24+108+18+24+2$
$=176$

Therefore total of $176$ words can be formed from the letters of the word $MISSISSIPPI$.