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A square within a triangle (Posted on 2016-07-20) Difficulty: 2 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
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solution Comment 1 of 1
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
-----------------------------------------------------------------
[(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).

  Posted by armando on 2016-07-20 09:58:47
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