Problem Solving

4 letters are randomly selected from PROPORTION.

a) Number of ways for selecting 3 alike and one different
b) Number of ways for selecting 2 alike and 2 others alike
c) Number of ways for selecting 2 alike and 2 other different
d) Number of ways for selecting all 4 different

Requesting to help me on solving this question.

P.S Answers are not available since I got the questions from a Handout

Given data
The frequency of each of the distinct alphabets in the word PROPORTION are:
P = 2, R = 2, O = 3, T = 1, I = 1, N = 1
Total no. of alphabets with repetitions = 10
Total no. of distinct alphabets = 6

a) No. of ways to select 3 alike = 1
No. of ways to select 1 different from 5 distinct letters = C(5,1) = 5
Total no. of ways = 5 x 1 = 5

b) No. of ways to select 2 alike = 3
Remaining letters are 2.
No. of ways to select the other 2 alike = 2.
Total no. of ways = 3 x 2 = 6

c) No. of ways to select 2 alike = 3
Remaining letters are two.
No. of ways you can select 2 distinct letters from 5
= C(5,2) = 10
Total no. of ways = 3 x 10 = 30


d)No. of ways of selecting all 4 different.
C(6,4) = C(6,2) = 21.

Experts, need your help on this one!