An interior number in Pascal's triangle is surrounded by 6 integers.
Prove that the product of these numbers is always a square of an integer.
Each entry in a Paschal triangle can be expressed as the combination C(n,m), being n the row and m the position from the left in the row.
So we have:
C(n-1,m-1) C(n-1,m)
C(n,m-1) C(n,m) C(n,m+1)
C(n+1,m) C(n+1,m+1)
Their product ahead of C(n,m) is:
[(n-1)!*n!*(n+1)!]^2
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[(m-1)!*m!*(m+1)!]^2*[(m-n)!*(m-n-1)!*(m-n+1)!]^2
which is always a square integer (as is the product of six integers).
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Posted by armando
on 2016-07-20 09:58:47 |
solution