A choir director must select a group of singers from among her 6 tenors and 8 basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let N be the number of groups that could be selected. What is the remainder when N is divisible by 100?

The required remainder when N is divided by 100 is 15.

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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.

Last Updated by eNotes Editorial on March 2, 2021
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