Adeck of 14 cards numbered 1 through 14 is dealt to ken and gunther so that each gets 7 cards. each player's score is the sum of his card values, and the player with the lower score wins. how many different winning scores are there?

The different winning scores possible are 25 (from 28 to 52).

Step-by-step explanation:

The minimum score obtained by a player will be 28.

since, we divide 1-7 numbered to one and 8-14 to the other player such that    one receives the maximum and the other receives the minimum.

 the least minimum possibility is :28 as one get the card as(1,2,3,4,5,6,7).

while the other player gets 77 sum and loses.

for increasing this score now we give the number 1,2,3,4,5,6,8 to one of the player so that they sum upto 29 and will get the minimum score and wins while the other player gets 76 sum with the help of numbers{9,10,11,12,13,14,7} and loses.    next possible minimum sum to win will be 30 by assigning 1,2,3,4,5,6,9 whereas the other gets the sum 75 and loses.continuing in the same way we see that the minimum sum obtained to win will be 52( by assigning 2,3,6,7,9,12,13) while the other gets the sum 53 and loses.

    Hence, there are total 25 winning scores possible.  (from 28 to 52 i.e. 52-28=25).

Agc and bgd are vertical angles