Question

In how many ways we can select $4$ letters from the letters of the word $MISSISSIPPI$?

Answer 1

Here we have to do it in cases. Let us first count letters and their repetition:—

$M—1$

$I—4$

$S—4$

$P—2$

Case I: When all the four letters are distinct:-

$P(4,4)=4!=24$

Case II: When two letters are repeated:-

the repeated letter can be selected from I,S,P i.e. $C(3,1)$ and the remaining two letters can be selected from 3 i.e. $C(3,2)$

$(3!/(2!\ast 1!\left)\right)\ast (3!/(2!\ast 1!)\ast (4!/2!)$

$(6\ast 6\ast 24)/(2\ast 2\ast 2)$

$=108$

Case III: When one is repeated twice and another one is repeated twice:-

$C(3,2)\ast 4!/(2!\ast 2!)$

$(3!\ast 4!)/(2!2!2!)$

$6\ast 24/(2\ast 2\ast 2)$

$=18$

Case IV: When one is repeated thrice and other is once:-

$C(2,1)\ast C(3,1)\ast 4!/3!$

$2\ast 3\ast 24/6$

$=24$

Case V: When one is repeated 4 times:-

$C(2,1)\ast 4!/4!$

$2\ast 4!/(4!)$

$=2$

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

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