#GilasPilipinas #Puso
To score in a basketball game, a player can either have
A) free throws (1 point each);
B) 2-point field goals;
C) 3-point field goals;
D) a combination of any of these.
In how many distinct ways can Gilas Pilipinas team captain Jimmy Alapag (jersey #4 in the picture) score a total of 25 points in a certain game? As an example, one way for him to score 25 points is to have 1 free throw, 6 2-point field goals and 4 3-point field goals.
(The image was taken from kalongkong.wordpress.com)
The answer is 65.
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1 solution
Let F , T , T h represent the number of free throws, 2-point field goals and 3-point field goals (respectively) made by Alapag. From the problem, F + 2 T + 3 T h = 2 5 ; F , T , T h ≥ 0 The RHS, 25, is odd, while 2 T of the LHS is always even for all T . So exactly one of F and T h is even.
Case 1 : F is even. Let F = 2 x , where x is a nonnegative integer. Then 2 x + 2 T + 3 T h = 2 5 ⟹ 2 ( x + T ) = 2 5 − 3 T h The LHS is even, so must be the RHS. So, T h must be odd. Since 2 ( x + T ) = 2 5 − 3 T h ≥ 0 , we must then have T h ≤ 3 2 5 ⟹ T h = 7 , 5 , 3 , 1
If T h = 7 , x + T = 2 , and so there are 3 nonnegative ordered pair solutions of ( x , T ) . For the cases T h = 5 , 3 , 1 , there are 6, 9, 12 ordered pair solutions, respectively.
So, for case 1, there are 12 + 9 + 6 + 3 = 30 ways.
Case 2 : T h is even. By the similar argument, you will find out that there are 5 + 4 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 = 35 ways for this case (Check this out! :) )
Therefore, there are 30 + 35 = 65 ways for Alapag to score 25 points in a cartain game.