Question

Prove that only 1422 different four letter words can be formed out of the letters of the word INEFFECTIVE.

Answer 1

Total $(3e,2f,2i$ rest 4 all different) i.e. 7 types.
(i) Al different ${{}^{6}P}_4=6\times 5\times 4\times 3=360$ .
(ii) Two different, two alike.
${{}^{2}C}_1\times {{}^{5}C}_2\times \frac{4!}{2!}=240$
(iii) 3 alike, 1 different
${{}^{1}C}_1\times {{}^{5}C}_1\times \frac{4!}{3!}=20$
(iv) 2 alike of one type and 2 alike of other type.
${{}^{2}C}_2\times \frac{4!}{2!2!}=6$.
$\therefore$ Total number of words $=360+240+20+6=626$.