Question

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

Answer 1

`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`.