Problem Solving : Permutation and combination Problem 1 : In how many ways can the letter of the word PROPORTION be arranged by taking 4 letters at a time?

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

Solution:

P R O P O R T I O N

below I've wrote letters without repeated words and then how many times particular word repeated
P R O T I N
P R O 
       O

Here (P, P) 2 P, (R, R) 2 R, and (O,O,O) 3 O

from the question here we gonna select 4 words hence we have the following chance to select those 4 words

Chance 1:   3 O and remaining 1 are different (That means (O,O,O)--> 3 word and (P,R,T,I,N) ---> 1 word from the 5)

Chance 2:  two of same words and other two of the  same words (here (O,O,O), (T,T) and (P,P) out of 3 pairs we gonna select 2 pair of words )

Chance 3: Two of same words and other two are different words (here (O,O,O), (T,T) and (P,P) out of 3 pairs we gonna select 1 pair of word and other 2 word from (R, O, I, N, T) words consider P,P has been selected pair  )

Chance 4: All 4 are different ( here P, R,  O, T, I , N out of 6 we gonna select 4 words)

Hence


Chance 1:  3C3 * 5C1 * (4! /3!) = 20 [3! for repeated words O,O,O]
Chance 2:  3C2 * (4! / (2!*2!)) = 18 [2! * 2! for repeated words of T,T and P,P]
Chance 3: 3C1*5C2 * (4! / 2!) = 360 [2! for repeated word T,T or P,P]
Chance 4: 6C4 * 4! = 360 [as usual]

finally add all the chances = 20 + 18 + 360 + 360 = 758 ways is the answer