Question

The number of ways in which we can select 5 letters of the word international is equal to

Answer 1

Answer:

256

Step-by-step explanation:

The Given word is

INTERNATIONAL

No. of letters in the word = 13

It contains, 3N, 2A, 2I, 2T, 1E, 1R, 1O, 1L

Case I

When all letters are different then from each we will take 1

Total arrangements = $\binom{8}{5}=\frac{8!}{5!3!} =56$

Case II

When three letters are same and two are different

Total arrangements = $\binom{3}{3}\times \binom{7}{2}=21$

Case III

When three letters are same and two are same

Total arrangements = $\binom{3}{3}\times \binom{3}{1}\times \binom{2}{2} =3$

Case IV

When three letters are different and other two are same

Total arrangements = $\binom{3}{1}\times \binom{2}{2}\times \binom{7}{3} +\binom{7}{3}=\binom{7}{3}\times 4=35\times 4=140$

Case V

When two letters and other two are same and one is different

Total arrangements = $\binom{3}{2}\times \binom{2}{2}\times \binom{2}{2}\times \binom{6}{1}+\binom{3}{1}\times \binom{6}{1}=18+18=36$

Total arrangements = 56 + 21 + 3 + 140 + 36 = 256