Question

The total number of ways of selecting $5$ letters from the letters $AAABBBCCDEF$ is-

• A. $3320$
• B. $75$
• C. $70$
• D. $72$

Answer 1

The correct option is D $72$

Selecting $5$ letters from AAABBBCCDEE

$A-3,B-3,C-2,D-1,E-1,F-1$

Case $\left(1\right)$

$3$ same, $2$ same of different kind

${}^2{C}_1\times {}^2{C}_1=4$

Case $\left(2\right)$

$3$ same, $2$ different

${}^2{C}_1\times {}^5{C}_2=20$

Case $\left(3\right)$

$2$ same, $2$ same of different kind, $1$ different.

${}^3{C}_2\times {}^4{C}_1=12$

Case $\left(4\right)$

$2$ same, $3$ different

${}^3{C}_1\times {}^5{C}_3=30$

Case $\left(5\right)$

$5$ different

${}^6{C}_5=6$

Total no. of ways $=72$.