There are 10 balls in a bag numbered from 1 to 10. Seven balls are selected at random. How many different ways are there of selecting the seven balls?

This is a combination problem. In mathematics, a combination is a selection of items without regard to the order in which they are selected.

The formula for combinations is:

C(n, r) = n! / [r!(n-r)!]

where:

• n is the total number of items,
• r is the number of items to choose,
• "!" denotes factorial, which means multiplying all positive integers up to that number.

In this case, n = 10 (the balls numbered 1 to 10) and r = 7 (the balls to be selected).

So, the number of ways to select 7 balls out of 10 is:

C(10, 7) = 10! / [7!(10-7)!] = 10! / [7!3!] = (1098) / (321) = 120

So, there are 120 different ways of selecting the seven balls.