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

Question

3 Deadly mistakes in Permutation and Combination - Exercise Question #4

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

Options 
 A. 12
B. 34
C. 42
D. 62
E. 94

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EgmatQuantExpert · Accepted Answer

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

aserghe1 · Answer

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

EgmatQuantExpert · Answer

Hey Everyone,

Official solution to the question has been posted.

Regards,
Ashutosh

HouseStark · Answer

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

Bambi2021 · Answer

I just learned something new. Zinnia is a flower!

samarpan.g28 · Answer

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).

Kinshook · Answer

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

Z - 1
I - 2
N - 2
A - 1

Words consisting of Z, A, I = 3! = 6
Words consisting of Z, A, N = 3! = 6
Words consisting of Z, 2Is = 3
Words consisting of Z, 2Ns = 3
Words consisting of A, 2Is = 3
Words consisting of A, 2Ns = 3
Words consisting of 2I, N = 3
Words consisting of 2N, I = 3
Words consisting of Z, I, N = 3! = 6
Words consisting of I, N, A = 3! = 6

Total 3-letter words = 4*6 + 6*3 = 24 + 18 = 42

IMO C

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

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

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