How Many Different 3-Letter Words Can Be Formed From The Word “ZINNIA” GMAT Problem Solving

Number of different 3 letter words = (No. of 3 letter words with two Is) + (No. of 3 letter words with two Ns) + (No. of 3 letter words with every letter distinct)
No. of 3 letter words with two Is: (No. of ways to select two Is) * (No. of ways to select a 3rd letter) * (No. of ways to place that 3rd letter)
No. of ways to select two Is = 1
No. of ways to select a 3rd letter = 3 (select 1 letter from Z,N,A)
No. of ways to place that 3rd letter = 3 (*I*I* — * marks where we can place the 3rd letter)
No. of 3 letter words with two Is = 1 * 3 * 3 = 9
No. of 3 letter words with two Ns = 9 (same logic as above)
No. of 3 letter words with every letter distinct: (No. of ways to arrange 3 out of the 4 distinct letters) = 4P3 = 24
Number of different 3 letter words = 9 + 9 + 24 = 42

Correct option: C

Z = 1
I = 2
N = 2
A = 1
Total = 6 letters
3-letter word in which all 3 letters are different
Total of 4 different letters (Z, I, N, A)
The number of ways to to select 3 letters = 4C3 = 4
These 3 selected letters can arrange themselves = 3! ways
3-letter word in which 2 letters are same and 1 letter is different
1 I___ = (3 * 3!) / 2! = 9
NN___ = (3 * 3!) / 2! = 9
Therefore, the number of 3-letter words formed = 4 * 3! + 9 + 9 = 24 + 18 = 42

Correct option: C

The word “ZINNIA” has 2-N, 2-I, 1-Z, and 1-A
3 letter words using 2 same letters and 1 different letter
We have two groups of similar letters
2-N
2-I
= Ways to select 1 group of similar letters from 2 groups * Ways to different from remaining 3-different letters * arranging of 3 letters
= 2C1 * 3C1 * 3!/2! = (2 * 3 * 6) / 2 = 18
3 letter words using all different letters
There are 4 different letters- Z, I, N, A
Ways to form 3-letter word from 4 different letters = 4P3 = 24
Total three letter words = 18 + 24 = 42

Correct option: C