Question

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

Answer 1

INDEPENDENT
11 letters (3N, 3E, 2D), I, P, T = 6 types.
We have to form 5 letters - word
I. All different = ${}^6{C}_5.5!=\left(6\right).5!=720$
II. 2 alike, 3 diff. = ${}^3{C}_1.{}^5{C}_3.\frac{5!}{2!}$
$=\left(3.10\right).60=1800$.
III. 3 alike, 2 diff. = ${}^2{C}_1.{}^5{C}_2.\frac{5!}{3!}$
$=\left(2.10\right).20=400$.
IV. 2 alike, 2 alike, 1 diff. = ${}^3{C}_2.{}^4{C}_1.\frac{5!}{2!2!}$
$=\left(3.4\right).30=360$.
V. 3 alike, 2 alike = ${}^2{C}_1.{}^2{C}_1.\frac{5!}{3!2!}$
$=\left(2.2\right).10=40$
Total selections = 6+30+20+12+4=72
Total words = 3320.