Question

How many distinct 3-letter permutations can you make from the letters in the word HEXAGON?

Answer 1

Answer:   210

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Explanation:

There are 7 unique letters in the word HEXAGON.

For the first slot, we have 7 choices. Then the next slot has 6 choices. Then the third slot has 5 choices. We count down by one each time we need to fill another slot.

Multiply out the values mentioned: 7*6*5 = 42*5 = 210

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Alternatively, you can use the nPr permutation formula with n = 7 and r = 3.

$_n P _r = \frac{n!}{(n-r)!}\\\\_7 P _3 = \frac{7!}{(7-3)!}\\\\_7 P _3 = \frac{7!}{4!}\\\\_7 P _3 = \frac{7*6*5*4*3*2*1}{4*3*2*1}\\\\_7 P _3 = 7*6*5\\\\_7 P _3 = \boldsymbol{210}$

We have the "7*6*5" show up again after the "4*3*2*1" portions cancel.