<h3>
Answer: 136</h3>
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Explanation:
- Player 1 has 16 people to kick the ball to
- Player 2 has 15 people to kick the ball to (since player 1 already is accounted for)
- Player 3 has 14 people to kick the ball to
And so on.
We'll have this countdown: 16, 15, 14, ..., 3, 2, 1, 0. By the time we get to the 17th player, they won't have anyone to kick the ball to. This is because they've already been counted in the previous scenarios. This is to avoid double-counting.
So the problem simply boils down to us adding up the numbers 0,1,2,...14,15,16.
This is an arithmetic sequence with first term a = 0 and common difference d = 1. The sum of the first n = 17 terms is...
Sn = (n/2)*(a + d(n-1))
S17 = (17/2)*(0 + 1(17-1))
S17 = (17/2)*(16)
S17 = 17*8
S17 = 136 is the final answer
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An alternative method:
Consider we have two slots A and B
For slot A, we have 17 choices because we have 17 players.
Then for slot B, we have 17-1 = 16 choices since we don't want to reselect whoever goes in slot A.
If order mattered, then we'd have 17*16 = 272 different permutations.
However, order doesn't matter. A sequence like {player1,player4} is identical to {player4,player1}. All that matters is the overall group rather than the order of it. Both of those examples say the same thing: they connect player 1 to player 4.
In short, the value 272 is too large and it double-counts the true answer. We need to divide by 2 to correct for this.
272/2 = 136
We arrive at the same answer as before. There are 136 kicks required so that each player kicks the ball to every other player.
