Question

What is the 40th row and the sum of all the numbers in it of pascals triangle?

Answer 1

See explanation...

Explanation:

The `40`th row of Pascal's triangle consists of the numbers:

`((39),(0)), ((39),(1)), ((39),(2)), ..., ((39),(38)), ((39),(39))`

where `((n),(k)) = (n!)/((n-k)! k!)`

The binomial theorem tells us that:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k`

So putting `a=b=1` we find that:

`sum_(k=0)^n ((n),(k)) = 2^n`

So the sum of the terms in the `40`th row of Pascal's triangle is:

`2^39 = 549755813888`