Prove that product of n consecutive numbers is divisible by n!

Question

Prove that product of n consecutive numbers is divisible by n! That is, If $ P = (a) (a+1)...(a + n -1)$ Then, $n!|P$

Prove it in two ways, one without induction and another with induction.

I tried to prove it with the help of Euclid Division Algorithm first because when I had countable terms like 3 terms of the form $(a)(a+1)(a+2)$ O could use it to prove that it was divisible by 6 or 3!. But I could not get anywhere.

Then I looked in the book which uses the principle of mathematical induction, which I think didn't used Induction correctly (pic attached)

~Thanks

Links to the book pages https://i.stack.imgur.com/VJjHU.jpg https://i.stack.imgur.com/6vwf3.jpg

Answer 1

The hint.

What is it $\binom{a+n-1}{n}$?