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!

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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.

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

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

(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18