How many words can be formed by taking 4 letters at a time out of the GMAT Problem Solving

Question: How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS?

      A.  756
      B. 1680
      C. 1698
      D. 2436
      E. 2454

Answer: E

Approach Solution (1):
The word contain 11 letters with repetition of letters.
We can choose 4 letters by
i) all the four letters are distinct
ii) Two distinct and two alike
iii) Two alike of one kind and two alike of other kind

i) 8 different letters are M, A, T, H, E, I, C, S
This can be chosen by ⁸C₄ × 4! = 1680

ii) Three pairs alike letters MM, AA, TT. One pair can be chosen by ³C₁ = 3 ways
Remaining two letters can be chosen by ⁷C₂
Each such group has 4 letters out of which 2 are alike.
Hence, it can be chosen by 4!/2!
Total number of ways = 3×⁷C₂×4!/2! = 756

iii) Three pairs of 2 alike letters. Two pairs can be chosen by ³C₂
³C₂ groups of 4 letters each.
There are 4 letters of which 2 alike of one kind and two alike of other kind in each group.
That can be arranged by 4!/2!× 1/2!
Total number of ways = ³C₂×4!/2!× 1/2! = 18
Total number of ways = 1680 + 756 + 18 = 2454
Correct option: E

Approach Solution (2):
4 different = ⁸P₄ = 1680
2 similar,2 different = 3 ×⁷C₂ × 4! 2! = 756
2 similar and 2 similar = 3 × 4!/2!*2! =18
Total = 2454
Correct option: E

Approach Solution (3):
M – 2
A – 2
T – 2
H – 1
E – 1
I – 1
C – 1
S – 1
Words of the form abcd = ⁸C_{4 *} 4! = 1680
Words of the form aabc = ³C₁ * ⁷C₂ * 4! / 2! = 3*21*12 = 756
Words of the form aabb = ³C₂ * 4! / 2! / 2! = 3*24/4 = 18
Number of words that can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS = 1680 + 756 + 18 = 2454
Correct option: E

“How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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