a deck of 14 cards numbered1through 14 is dealt 7 each to two players. Each player score is the sum of their cards. lowest score wins. how many different winning scores are possible?

Guest Dec 14, 2018

#1**+1 **

Here's my best attempt :

When each player has an even number of cards, there will only be 1 equal sum between the players....to see this....let's suppose that we have cards 1 - 4

The possible outcomes between two players are

(1, 2) < (3,4)

(1, 3) < (2,4)

(2, 3) = (1,4)

Note that the number of "winning" scores is C(4,2)/ 2 - 1 = 3 - 1 = 2

When each player has an odd number, n, of cards, the number of winning scores is just C(2n, n) / 2

Suppose we have cards 1- 6

The outcomes are

(1,2, 3) < (4, 5, 6)

(1,2,4) < (3, 5, 6)

(1, 2, 5) < (3, 4, 6)

(1, 2, 6) < (3, 4, 5)

(1, 3, 4) < ( 2, 5, 6)

(1,3, 5) < ( 2, 4, 6)

(1, 3, 6) < ( 2, 4, 5)

(1, 4, 5) > (2, 3, 6)

(1, 4, 6) >( 2, 3, 5)

(1, 5, 6) > ( 2, 3, 4)

Note that the number of winninig scores is just C(6,3)/2 = 20/2 = 10

This implies....that.....for 14 cards....each player holds 7 and the number of winning hands =

C(14, 7) / 2 = 1716 winning hands

CPhill Dec 14, 2018