A choir director has to select a group of singers from among her 6 tenors and 8 basses. The difference between the numbers of tenors and basses must be a multiple of 4, and the group must have at least one singer. If the number of groups that can be selected is N, the remainder has to be determined when N is divided by 100.
If the selected group has x tenors and y basses,
x - y = 4k, where k is an integer and can be negative, positive or zero. As there has to be at least 1 singer in the group, both x and y cannot be 0.
If x = 0, y can be any of the numbers (4,8)
If x = 1, y can be (1, 5)
If x = 2, y can be (2, 6)
If x = 3, y can be (3, 7)
If x = 4, y can be (0, 4, 8)
If x = 5, y can be (1, 5)
If x = 6, y can be (2, 6)
This gives the total number of groups that can be created as 2+2+2+2+3+2+2 = 15.
The remainder when 15 is divided by 100 is 15.