The pascal triangle can be used to solve counting problems.
Let's say we have 6 students and we need to choose one student to do a choir. There are six ways to make the single choice.
If we need two students to do the play, we have 6 choices for the first student, and 5 for the second to make 30 choices. Out of the 30 choices, we have made each choice twice, because we could have chosen students A & B and another time B & A. So there are really 15 choices.
If we continue this way, we would come up with the numbers 6, 15, 20, 15, 6 for the number of ways of choosing 1, 2, 3, 4 and 5 students respectively.
If we add the single choices of choosing no students or all of the students, we have the complete 6th line of the Pascal's triangle of:
1 6 15 20 15 6 1