Question

Find the number of combinations and permutations of $4$ letters taken from the word $EXAMINATION$.

Answer 1

$4$ letter words from EXAMINATION

There are $8$ different letters "examination"

namely,

E,X,A,M,I,N,T,O

1, 1, 2, 1, 2,2,1, 1

Case (i):- all $4$ letters of the word are different.

no. of ways ${\Rightarrow }^8{C}_4\times 4!\Rightarrow \frac{8!}{4!4!}\times 4!\Rightarrow \frac{8!}{4!}\Rightarrow 1680$

Case (ii):- $2$ letters are same and $2$ letters are different we select are of $A$,$1$ and $N$ for $2$ same letters

No. of ways ${\Rightarrow }^3{C}_1{+}^7{C}_2\times \frac{4!}{2!}\Rightarrow 3\times \frac{7!}{2!\times 5!}\times \frac{4!}{2!}=756$

Case (iii)= $2$ letters same and other $2$ letter are also same

No. of ways ${\Rightarrow }^3{C}_2\times \frac{4!}{2!2!}\Rightarrow 3\times \frac{4!}{2!2!}\Rightarrow 18$

$Totalnoof4letterswords\Rightarrow 1680+756+18\Rightarrow 2459$