Find the number of (i) combinations. (ii) permutations of four letters taken from the word EXAMINATIONS.

Explanation

In the word 'CALCUTTA',A,C and T occur twice each and L,U occur one each. The classification of 4 letters cab be as follows:
(i) Two similar letters of 1st type and two similar letters of 2nd type No. of ways of selecting 2 letters from 3 letters `.^(3)C_(2) =3` and their arrangements `= (4!)/(2!2!) = 6`
`:.` Required permutations `= 3 xx 6 = 18`
(ii) Two similar letters of one type and other two different letters No. of ways of selecting 2 similar letters and 2 different letters `= .^(3)C_(1) xx^(4)C_(2) = 3 xx 6 = 18`
and their arrangements `= (4!)/(2!) = 12`
`:.` Required permutations `= 18 xx 12 = 216`
(iii) ALl 4 letters are different
No. of ways of selecting 4 letters `= .^(5)C_(4) = 5` and their arrangements `= .^(4)Pa = 24`
`:.` Required permutations `= 5 xx 24 = 120` Now,
Totak combinations `= 3 +18 +5 = 26` and total temperatures `= 18 +216 +120 = 354`.