The 4th number in the 32nd row of pascals triangle is the sum of how many triangular numbers?

1 Answer
Nov 5, 2017

#30#

Explanation:

First, we must find the fourth number of the thirty-second row of Pascals. That can be found with:

#""_32C_3 = (32*31*30)/(3*2*1)#

However, I won't find the number just yet. In a minute, you'll see why.

Next, we need to know the formula for the sum of triangular numbers. This is:

#(x*(x+1)*(x+2))/6# Inductive proof found here

Now, we need to find a value of #x# such that:

#(x*(x+1)*(x+2))/6=(32*31*30)/(3*2*1)#

However, I can simply reorder the right hand side to look like this:

#(x*(x+1)*(x+2))/6 =#

#(30*(30+1)*(30+2))/6#

Now, it's easy to notice that #x=30#.