How many different 3-letter words can be formed from the word “ZINNIA”

Solution

Given:
• We are given a 6-letter word- ‘ZINNIA’.

To find:
• We need to find the number of 3-letter different words using the letters of ‘ZINNIA’.

Approach and Working:

ZINNIA is a letter word having two N, two I, one Z, and one A.

Thus, different three letter words can be formed in two ways:

• Using 2 same and 1 different Or,
• Using all different letters.

o Using Z, I, and A.

3-letter words using 2 same and 1 different letters:

• We have 2 groups of similar letters

o Two N or,
o Two I.
• = Ways to select 1 group of similar letters from 2 groups * ways to different from remaining 3-different letters * arranging all three letters

• = \(^2C_1\) * \(^3C_1\)*\(\frac{3!}{2!}\)
• =2*3*6*/2=18

3-letter words using all different letters:

• We have 4 different letters – Z, I, N, A.
• Thus, ways to form 3-letter word from 4 different letters=\(^4P_3\)=24

Thus, Total three letter words= 18+24=42.
Hence, the correct answer is option C.

Answer: C
_________________

Number of different 3 letter words = (# of 3 letter words with two Is) + (# of 3 letter words with two Ns) + (# of 3 letter words with every letter distinct)

# of 3 letter words with two Is: (# of ways to select two Is) * (# of ways to select a 3rd letter) * (# of ways to place that 3rd letter)
# of ways to select two Is = 1
# of ways to select a 3rd letter = 3 (select 1 letter from Z,N,A)
# of ways to place that 3rd letter = 3 (*I*I* --> * marks where we can place the 3rd letter)
# of 3 letter words with two Is = 1*3*3 = 9

# of 3 letter words with two Ns = 9 (same logic as above)

# of 3 letter words with every letter distinct: (# of ways to arrange 3 out of the 4 distinct letters) = 4P3 = 24

Number of different 3 letter words = 9+9+24=42

Answer: C

Hey Everyone,

Official solution to the question has been posted.

Regards,
Ashutosh
_________________

ZINNIA

Z=1
I=2
N=2
A=1
TOTAL= 6 LETTERS

3-LETTER WORD IN WHICH ALL 3 LETTER ARE DIFFERENT

TOTAL OF 4 DIFFERENT LETTERS(Z, I, N, A), THE NO. OF WAYS 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

I I __ =\( 3*3!/2! = 9\)

N N __= \(3*3!/2! = 9\)

THEREFORE, THE NUMBER OF 3-LETTER WORDS FORMED= \(4*3!+9+9=24+18=42\)

ANSWER C

I just learned something new. Zinnia is a flower!

ZINNIA=Z(II)(NN)A
Case 1: All different letters: 4C3*3!=24.
Case 2: Two letters are the same, one letter is different: 2C1*3C1*(3!/2!)=18.
Combining, 24+18=42(C).