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.

Incidentally, you don't even have to choose the same two rows from the triangle.

Here is the 3rd times the 4th row with an offset of 1:

1    3    3    1    0    0
0    1    4    6    4    1
0 +  3 + 12 +  6 +  0 +  0 = 21 which is in row 7.

Posted by Jer on 2011-10-05 10:53:14