Question

How many different words can be formed by taking 4 letters at a time out of the letters of the word EXPRESSION?

Answer 1

EXPRESSION
$10,(2E,2S,$ rest all six different), in all 8 types.
1. All different ${{}^{8}C}_4$ selection
Arrangement ${{}^{8}C}_4.4!$
2. 2 alike, 2 different ${{}^{2}C}_1.{{}^{7}C}_2$ selection
Arrangements ${{}^{2}C}_1.{{}^{7}C}_2.\frac{4!}{2!}$
3. 2 alike, 2 alike ${{}^{2}C}_2=1$ selection
Arrangements $=\frac{4!}{2!2!}$
Total no. of selection is $70+42+1=113$.

Total arrangements $=1680+504+6=2190$.