In how many ways can the letters of the word TERRORISTS be arranged by taking 4 letters at a time?

Solution

There are 10 letters in the given word TERRORISTS and we have 10 letters of 6 different kinds viz, (T,T,T), (I,I), (S,S), E, O, R For a group of four letters, we have four cases (i) Three alike and one different (ii) Two alike and two other alike (iii) Two alike and other two different (iv) All four different In case (i), the number of selections = 3C3 × 5C1 = 5 In case (ii), the number of selections = 3C2 = 3 [Since, we can select two pairs out of 3 pairs (T,T) (I,I) (S,S)] In case (iii), the number of selections = 3C1 × 5C2 = 30 [since, we can select one of 3 pairs and then two from the remaining 5 letters say, I,S,E,O,R] In case (iv), the number of selections = 6C4 = 15 [since, we can select 4 different letters from 6 letters T, I,S,E,O,R ] In case (i), the number of arrangements = 5 ×4!/3! = 20 In case (ii), the number of arrangements = 3 ×4!/(2! ×2!) = 18 In case (iii), the number of arrangements = 30 ×4!/2! = 360 In case (iv), the number of arrangements = 15 × 4!= 360 Hence, the required number of arrangements = 20 + 18 + 360 + 360 = 758