Question

A 8 year old boy has 6 different toys and wants to put them all in a straight line.
In how many ways can this be done?

I would appreciate step by step, as I have no clue on how to solve. Thanks!

Answer 1

Answer:  720

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Explanation:

The number 8 from "8 year old boy" can be completely ignored. In my opinion, this is an (un)intentional distraction on your teacher's part.

There are 6 toys to arrange. The order is important.

• For the first slot, there are 6 choices.
• Then the second slot has 5 choices (we cannot have a toy occupy more than one slot at a time).
• The third slot has 4 choices, and so on.

We have this countdown: 6,5,4,3,2,1

Those values multiply out to 6*5*4*3*2*1 = 720

There are 720 ways to arrange the 6 different toys. Order matters.

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An alternative approach is to use the nPr permutation formula with n = 6 and r = 6. We use a permutation because order matters.

The nPr formula is

$_{n} P _{r} = \frac{n!}{(n-r)!}$

where the exclamation marks indicate factorial. For example, 6! = 6*5*4*3*2*1 = 720.