Take a row of Pascals Triangle. Write a second copy of the row below it but with an offset. Multiply each pair of numbers and then sum these products. (A missing number is considered to be zero.)
The result is number further down the Triangle.
Can you explain or prove the result?
Example with the 4th row and offset by 1:
1 4 6 4 1 0
0 1 4 6 4 1
0 + 4 + 24 + 24 + 4 + 0 = 56 which is in row 8.
(In reply to Happy 35th anniversary (spoiler)
by Steve Herman)
Another isomorphic example for the two equalities Steve proved:
1st: Selecting a team of n persons out of 2n people(n of them males, n females).<br>
2nd: Selecting a team of n-k persons out of 2n people (n of them males, n females).