Factorials are unnecessary

Algebra Level 1

$1^{10}\times 2^9\times 3^8\times \cdots \times 10^{1} , \qquad 1!\times2!\times3!\times \cdots \times10!$ Which of the numbers above is larger?

Details and Assumptions:

$n! = 1\times2\times3\times\cdots \times n$ .

1^{10}\times 2^9\times 3^8\times \cdots \times 10^{1}

1!\times2!\times3!\times \cdots \times10!

They are equal

1 solution

Anandmay Patel
Dec 4, 2016

This is a great question.

Notice that the expression $1!\times 2!\times 3!\times \dots \times 10!$ supplies $\textbf{ten}$ 1s, $\textbf{nine}$ 2s, $\textbf{eight}$ 3s and so on...... till $\textbf{one}$ 10.

See the reason why?

For example,1! contributes a 1,2! contributes another one,3! contributes another one.......

So,total ten 1s.

Again,starting with 2!,2! contributes a 2,3! contributes a 2.........with a total of nine 2s.

Like-wise we can continue,to form an equivalent expression for $1!\times 2!\times 3!\times \dots\times 10!$ as $1^{10}\times 2^9.......\times 10^1$ .

So both the expressions are $\textit{equal}$ .

I like your conversation-styled solution. It makes me easy to follow!

Christopher Boo - 4 years, 6 months ago

Thanks.I am glad that it helps the community to understand in ease and joy!

Anandmay Patel - 4 years, 6 months ago

Factorials are unnecessary

1 solution

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