Number of arrangements of the word "MAMMAL" where M is not together

Question

This is in reference to this question. Letter Arrangement with Permutations

_A_A_L_

IF M is not together, then M can go into 4 distinct places (denoted by the underscores above). So the number of ways the Ms can be places is $\binom{3 + 4 -1}{3}$ and the letters A A L can be arranged in $\frac{3!}{2!}$ ways. So the total number of arrangements is $\binom{6}{3} \times 3 = 60$

Is this the correct way of doing this question?

Choose 3 of 4 places, not 3 of 3+4-1 places. – Graham Kemp Nov 25 '14 at 03:49 — Graham Kemp, Nov 25 '14 at 03:49

score 3 · Accepted Answer · answered Nov 25 '14 at 03:42

No, since Ms can't be together, each slot can only contain a maximum of one M, so the number of ways the Ms can be arranged in the slots is just the number of ways to choose which slot is empty. There are just four possibilities, not $6\choose3$. The number of ways to arrange AAL is 3 as you say. The total number of arrangements is $3\times4=12$.

Number of arrangements of the word "MAMMAL" where M is not together

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