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June 1st, 2014, 11:01 PM   #1
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How to prove this factorial problem?

I have a question about factorials and provide a few examples below.

(ex 1) 6! = 6*5*4*3*2*1

(ex 2) 9! = 9*8*7*6*5*4*3*2*1

(ex 3) 11! = 11*10*9*8*7*6*5*4*3*2*1

(ex 4) n! = n*(n-1)*(n-2)*(n-3)...3*2*1

So, the r th term in each factorial above is n - r + 1.

For example, the 4th term in (ex 1) is 6 - 4 + 1= 3, the 5 th term in (ex 2) is
9 - 5 + 1 =5 and the 3 rd in (ex 3) is 11 - 3 + 1 = 9.

I figured out the r th term = n - r + 1 by finding a pattern in examples 1 - 3 above.

Can someone show me how to prove n - r + 1 for the r th term. Thanks.
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June 2nd, 2014, 12:15 AM   #2
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We have 1 <= r <= n
The first factor is n. The second factor is the first factor - 1.
The third factor is the second factor - 1 is the first factor - 2, and so on.
The r-th factor in the product is the first factor - (r - 1) = n - (r - 1) = n - r + 1. See?
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