Question

Find the total number of ways of selecting five letters from the word INDEPENDENT.

Answer 1

word INDEPENDENT has $2Ds,3Es,3Ns$ and $I,P,T$
Hence the 5 numbers can be selected as follows:
3 alike ,2 different =$\left(\begin{array}{c}2\\ 1\end{array}\right)\times \left(\begin{array}{c}5\\ 2\end{array}\right)=20$
3 alike ,2 other alike=$\left(\begin{array}{c}2\\ 1\end{array}\right)\times \left(\begin{array}{c}2\\ 1\end{array}\right)=4$

2 alike ,3 different=$\left(\begin{array}{c}3\\ 1\end{array}\right)\times \left(\begin{array}{c}5\\ 3\end{array}\right)=30$
2 alike ,2 other alike and 1 different=$\left(\begin{array}{c}3\\ 2\end{array}\right)\times \left(\begin{array}{c}4\\ 1\end{array}\right)=12$
All different $=\left(\begin{array}{c}6\\ 5\end{array}\right)=6$
Therefore, total no. of ways of selection $=(20+4+30+12+6)=72$