Fundamental principle of counting

Question

I have this question,

The number of different words that can be formed using all the letters of the word “MATHEMATICS” are?

I solved using the logic that if there are total 11 places so at all the 11 places all the 11 words can come, which gives me 11 to the power 11which is then divided by $2!*2!*2!$ due to the words M, T and A appearing 3 times.

But the actual solution is $11!/(2!*2!*2!)$, why did they consider “not repetition” ?

Is the usual interpretation of "using all the letters of the word". Example for SE: https://math.stackexchange.com/questions/483277/how-many-different-words-can-be-formed-using-all-the-letters-of-the-word-googolp. — Martín-Blas Pérez Pinilla, Mar 21 '18 at 09:35
Even if the problem were asking with repetition (which it isn't), your answer would be wrong. Clearly $11^{11}/2^3$ is not an integer. If it were with repetition, the answer would be $8^{11}$ (assuming you mean strings with $11$ letters taken from the alphabet {M,A,T,H,E,I,C,S}).. — anon, Mar 21 '18 at 09:39

score 2 · Accepted Answer · answered Mar 21 '18 at 09:32

2

Your power means repetition. $11^{11}$ is the cardinal of a cartesian product $X\times\cdots\times X$ (11 times). of a set $X$ with 11 elements.

answered Mar 21 '18 at 09:32

Martín-Blas Pérez Pinilla

41,546
4
46
89

Yes, because according to me if the repetition is allowed I can place all 11 letters in all 11 places then why does the actual answer doesn’t takes repetition into account ? – Gagan Batra Mar 21 '18 at 09:34
@GaganBatra, see my comment. – Martín-Blas Pérez Pinilla Mar 21 '18 at 09:35
@GaganBatra, and if repetition is allowed, the power in any case will be $8^8$ (or $11^8$). – Martín-Blas Pérez Pinilla Mar 21 '18 at 09:37
You mean $8^{11}$ if repetition were allowed (and the word had $11$ letters). – anon Mar 21 '18 at 09:42
@anon, right... – Martín-Blas Pérez Pinilla Mar 21 '18 at 10:50
But why can’t I take my answer as 8^11, because there is nothing mentioned about repeatation. – Gagan Batra Mar 23 '18 at 19:18
1

@GaganBatra, again: because is the usual interpretation. – Martín-Blas Pérez Pinilla Mar 23 '18 at 19:24

Fundamental principle of counting

1 Answers1