find the number of ways in which a] a selection b] an arrangement of 4 letters can be made using the word PROPORTION ???
- PL ANSWER..
The word is PROPORTION
Number of P's - 2
Number of R's - 2
Number of O's - 3
Number of T's - 1
Number of I's - 1
Number of N's - 1
Let us first find the number of selections.
Case 1: All four letters distinct - Number of ways will be 6C4 = 15
Case 2: Three letters same and one letter distinct - Number of ways = 5C1 = 5. In this case we will have three R's and any one other letter.
Case 3: Two letters of one type and the other two letters of other type - Number of ways = 3C2= 3. In this we need to choose two letters out of P, R and O.
Case 4: Two letters same and the other two letters different - Number of ways = 3C1 x 5C2= 3 x 10 = 30. In this we need to select one out of R, O and T (they will be occurring two times) and the other two letters distinct.
Total number of selections = 15 + 5 + 3 + 30 = 53.
Let us now find the number of arrangements. We will take the cases separately (as discussed above)
Case 1: 15 x 4! ways = 15 x 24 = 360
Case 2: 5 x (4!/3!) = 20
Case 3: 3 x (4!/(2! x 2!)) = 18
Case 4: 30 x (4!/2!) = 360
Total arrangements = 360 + 20 + 18 + 360 = 758