If each child gives a farewell gift to every other child in a nursery, then the total number of gifts would be 650. How many children are there in the nursery?

This problem can be solved using the formula for combinations in a set, which is n(n-1)/2, where n is the number of items in the set. In this case, the set is the number of children in the nursery, and we want to find out how many combinations of 2 (i.e., pairs of children) can be made.

Given that the total number of gifts is 650, we can set up the equation:

n(n-1)/2 = 650

Solving for n, we get:

n^2 - n - 1300 = 0

This is a quadratic equation, and we can solve it using the quadratic formula, which is:

n = [-b ± sqrt(b^2 - 4ac)] / 2a

In this equation, a = 1, b = -1, and c = -1300. Plugging these values in, we get:

n = [1 ± sqrt((1)^2 - 41(-1300))] / 2*1 n = [1 ± sqrt(1 + 5200)] / 2 n = [1 ± sqrt(5201)] / 2

The square root of 5201 is approximately 72.11. Since we can't have a fraction of a child, we round this to the nearest whole number. So, the possible values for n are:

n = (1 + 72) / 2 = 36.5 n = (1 - 72) / 2 = -35.5

Since we can't have a negative number of children, the only possible solution is n = 36.5. But since we can't have half a child, we round this up to the nearest whole number, which is 37.

So, there are 37 children in the nursery.