Question

Show that only 286, 4 - letter words can be formed out of the letter of the word 'infinite'

Answer 1

8, (3i, 2n 3 different) 5 types.
We have from 4 letter words.
I. All different $={}^5{C}_4.4!$ or ${}^5{P}_4=120$
II. 2 alike, 2 diff. $={}^2{C}_1{.}^4{C}_2.\frac{4!}{2!}=144$
III. 2 alike, 2 diff. $={}^2{C}_2.\frac{4!}{2!2!}=6$
IV. 3 alike, 1 diff. $={}^1{C}_1{.}^4{C}_1.\frac{4!}{3!}=16$
Total $=120+144+6+16=286$.