Question

How many words can be formed by taking $4$ letters at a time of the letters of word $MATHEMATICS$

• A. $2234$
• B. $2542$
• C. $2346$
• D. $2454$

Answer 1

The correct option is D $2454$

Given words $MATHEMATICS$

for our simplicity we can write it as

$MATHEICS$

$MAT$

a) words of type $ABCD$

(All four letters are different)

(4 letters are chosen from $MATHEICS$ and are arranged)

$=8_{c_4}\times 4!=1680$

b) words of type $AABC$

(2 are alike, 2 are different)

(1 pair of letters is selected from M, A, T and 2 letters

are chosen from the remaining 7 letters and are arranged)

$=3_{c_1}\times 7_{c_2}\times \frac{4!}{2!}=756$

c) words of type $AABB$

(2 are alike of 1 kind, 2 are alike of another kind)

(2 pair of letter are chosen from M, A, T and arranged)

$=3_{c_2}\times \frac{4!}{2!2!}=18$

Total no. of word $=1680+756+18$

$=2456$