How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
Choices
A) 2154 B) 2454. C) 2254 D) 2354 E) 1245
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).   Selected 4 letters can have following 3 patterns:   1. abcd - all 4 letters are different:   8P4=16808 (choosing 4 distinct letters out of 8, when order matters) or 8C4∗4!=16808 (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);   2. aabb - from 4 letters 2 are the same and other 2 are also the same:   3C2∗4!/2!2!=18 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by 4!/2!2! to get different arrangements (for example MMAA can be arranged in 4!/2!2! # of ways);   3. aabc - from 4 letters 2 are the same and other 2 are different:   3C1∗7C2∗4!/2!=756 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by 4!/2!to get different arrangements (for example MMIC can be arranged in 4!/2!# of ways).   1680+18+756= 
<gadget id="calculator">1680+18+756</gadget>
<output>2_454</output>
2454   Answer: 2454.
<result>B</result>
How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.