There are 7 vacant places in a company and 10 people applied. We need to find out how many ways these seats can be filled.



Before we start, it is important to understand the difference between permutation and combination. In permutation, the order of selection is important, whereas in combination, the order of selection is not important.

For example, if we have three letters A, B, and C and we need to select two of them, then in permutation, we will have six possible outcomes (AB, AC, BA, BC, CA, CB), whereas in combination, we will have only three possible outcomes (AB, AC, BC).



In this case, we need to select 7 people out of 10 to fill the vacant places. Since the order of selection is not important, we will use combination to calculate the number of ways seats can be filled.

We can use the formula for combination:

nCr = n! / (r! * (n-r)!)

where n is the total number of items, r is the number of items to be selected, and ! denotes factorial (product of all positive integers up to that number).

In this case, n = 10 and r = 7.

So, the number of ways seats can be filled is:

10C7 = 10! / (7! * (10-7)!) = 10! / (7! * 3!) = (10*9*8) / (3*2*1) = 120

Therefore, there are 120 ways seats can be filled in the company.