Question

If a “word” is any arrangement of 3 different letters, how many 3-letter “words” can be formed from the letters B, O, N, and K, where each letter is to be used once?

Answer 1

Answer:  24

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Explanation:

We have 3 slots, which I'll call slot 1, slot 2, slot 3.

• For slot 1, we have 4 choices to pick from
• Then slot 2 has 3 choices since we can't reuse a letter
• Finally, slot 3 has 2 choices. We count our way down each time we move to the next slot.

Once all the slots are accounted for, we multiply the values mentioned: 4*3*2 = 12*2 = 24.

There are 24 different three-letter "words" that can be formed where any letter selected cannot be reused.

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A slightly different approach:

Order matters because a "word" like BON is different from a "word" like BNO.

Since order matters, we'll use a permutation

We have n = 4 letters to pick from and r = 3 slots to fill.

Apply those values into the nPr permutation formula

$_{n} P_{r} = \frac{n!}{(n-r)!}\\\\_{4} P_{3} = \frac{4!}{(4-3)!}\\\\_{4} P_{3} = \frac{4!}{1!}\\\\_{4} P_{3} = \frac{4*3*2*1}{1}\\\\_{4} P_{3} = \frac{24}{1}\\\\_{4} P_{3} = 24$

We end up with the same result as before.