Suppose a peewee basketball team scores 24 points in a game. How many combinations of free throws (one point each), two-pointers, and three- pointers are possible?

Answer 1

All one pointers, 24 of them

1 3 pointer plus 21 one pointers

2 3 pointers plus 18 one pointers

3 3 pointers plus 15 one pointers

4 3 pointers plus 12 one pointers

5 3 pointers plus 9 one pointers

6 3 pointers plus 6 one pointers

7 3 pointers plus 3 one pointers

8 3 pointers, all 3 pointers

9 subtotal so far

then

all 2 pointers, 12 of them, for 10 subtotal

then add in mixtures of twos and ones

11 2 pointers 2 one pointers

10 2 pointers 4 one pointers

9 2 pointers 6 one pointers

8 two's, 8 one's

7 two's 10 one's

6 two's 12 ones'

5 two's 14 ones

4 two's 16 ones

3 two's 18 ones

2 two's 20 ones

1 two, 22 ones

11 more for up to 21 ways, so far

then add in mixtures of just twos and threes

2 threes, 9 twos

4 threes, 6 twos

6 threes, 3 twos

that's 3 more, up to 24 ways subtotal

that covers, all ones, all, twos, all threes

and mixtures of ones and threes and ones and twos, and twos and threes

what's left is a mixture of all 3, ones,twos and threes

1 three, 1 two, 19 ones

1 three 2 twos, 17 ones

1 three 3 twos, 15 ones

1 three 4 tows 13 ones

1 three, 5 tows 11 ones

1 three 6 twos 9 ones

1 three 7 twos 7 ones

1 three 8 tows 5 ones

1 three 9 twos 3 ones

1 three 10 twos 1 one

2 threes 1 two 16 ones

2 threes 2 twos 14 ones

2 threes 3 twos 12 ones

2 threes 4 twos 10 ones

2 threes 5 twos 8 ones

2 threes 6 twos 6 ones

2 threes 7 twos 4 ones

2 threes 8 twos 2 twos

3 threes 1 two 16 ones

3 threes 2 twos 14 ones

3 threes 3 twos 12 ones

3 threes 4 twos 10 ones

3 threes 5 twos 8 ones

3 threes 6 twos 6 ones

3 threes 7 twos 4 ones

3 threes 8 twos 2 ones

4 threes 1 two 11 ones

4 threes 2 twos 9 ones

4 threes 3 twos 7 ones

4 three 4 twos 5 ones

4 threes 5 twos 3 ones

4 threes 6 twos 1 one

5 threes 1 two 7 ones

5 threes 2 twos 5 ones

5 threes 3 twos 3 ones

5 threes 4 twos 1 one

6 threes 1 two 4 ones

6 threes 2 twos 2 ones

7 threes 1 two 1 one

that add another 39, to 24

for a grand total of 63 different ways

or if you just wanted combinations of all 3, then 39 different ways