[Math] How many distinct permutations can you make with the the letters in word APPLE, AASSB

combinatoricspermutations

How many words can you make by permuting the letters of the word APPLE?

My attempt

$$\binom{5}{1}\binom{4}{2}\binom{2}{1}\binom{1}{1} = 60$$

What is the number of words you can make by permuting the letters of AAASB?

My attempt

$$\binom{5}{3}\binom{2}{1}\binom{1}{1} = 20$$

My question is why am I using the combination formula to find permutations? I'm having a mental block, or not seeing it clearly…… could someone clarify?

Best Answer

For the number of permutations of $n$ letters, where one or more letter is repeated among the $n$ letters, we can use the multinomial theorem:

The multinomial coefficient is also the number of distinct ways to permute a multiset of $n$ elements, and $k_i$ are the multiplicities of each of the distinct elements.

For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has $1$ M, $4$ Is, $4$ Ss, and $2$ Ps and has $11$ characters all together (where $11$ is equal to the sum the multiplicities) is given by:

$$\binom{11}{1, 4, 4, 2} = \frac{11!}{1!\cdot 4! \cdot 4!\cdot 2!}$$

What we have in both cases is a word with five characters, each with characters that repeat.

The letters in APPLE have only the repetition of the letter P, and the other letters appear only once.. Then the number of permutations of APPLE is equal to $$\frac{5!}{1!1!1!2!} = \frac{5!}{2!} = 5\cdot 4\cdot 3 = 60 \text{ distinct permutations of the letters in APPLE}$$ where the denominator is the product of the number of occurrences of each letter in the original word (and they sum to $n = 5$).

In the case of AASSB, we have the letter A with two occurrences, the letter S with two occurrences, and the letter B with one occurrence:

The number of permutations of this number is given by $$\frac{5!}{2!2!1!} = \frac{5\cdot 4\cdot 3}{2} = 30 \text{ distinct permutations of the letters of AASSB}.$$

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