Physics and Maths Tutoring from Michael A Smith.

Preamble

This seems like a hard calculation, but with a little thought this can be done with pencil and paper.

Warm-up:

Factorial

The Problem:

How many weeks are 10! seconds?

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Solution:

In order to find the number of weeks, 10! seconds needs to be divided by the number of seconds in a week. That is,

\frac{10!}{7 \times 24 \times 60 \times 60}

Expanding the factorial ten.

\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{7 \times 24 \times 60 \times 60}

Notice that the sevens cancel, giving,

\frac{10 \times 9 \times 8 \times 6 \times 5 \times 4 \times 3 \times 2}{24 \times 60 \times 60}

The one has gone too as multiplying by one makes no change. Now the three and eight in the numerator cancel with twenty four in the denominator, giving,

\frac{10 \times 9 \times 6 \times 5 \times 4 \times 2}{60 \times 60}

Now the ten and six in the numerator cancel with one of sixties in the denominator, giving,

\frac{9 \times 5 \times 4 \times 2}{60}

Finally, a factor of three from the nine and the five and four cancel with the final sixty, giving,

3 \times 2 = 6

So, there are exactly 6 week in 10! seconds.

Binomial Expansion

Permutations and Combinations
(care of revisionmaths.com)

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