How Many Different Cricket Teams Can Be Chosen From A List Of 17 Players?
28 October 2009
5 Comments
How many different cricket teams can be chosen from a list of 17 players? My book says 12,376 but doesn’t show me how to do it.
How do I do it without using a calculator, please? Many thanks. I know people today use calculators but this book was written years ago so how did people do it without them? Thanks.
Well, first, let’s consider how many different ways the 17 players can stand in a line.
There are 17 possible players who could be first in line. For each such choice, there are 16 possible players who could be second in line, giving 17 * 16 choices for the first two spots. For each of these choices, there are 15 possible choices for the third position, which makes 17 * 16 * 15 ways to pick people in the first three spots. By continuing in this manner, we can see that the number of ways to arrange the 17 players in a line is 17 * 16 * 15 * … * 3 * 2 * 1. This is usually written as 17! (the exclamation point is a symbol for “factorial”–it’s read as “17 factorial”).
So, how many teams of 11 players can we make?
Well, we know there are 17! ways to line the players up. For each of these, we could just say “The first 11 players in line make up one possible team.” But we don’t want to count, say, the arrangement
1, 2, 3, 4, 5, 6, …, 17
separately from
2, 1, 3, 4, 5, 6, …, 17,
because they have the same first 11 spots, and so would determine the same cricket team. Given a particular arrangement, there are 11! ways to arrange the first 11 players which would result in the same team, so we have to divide 17! by 11!.
Similarly, it doesn’t matter how the last 6 players are arranged–if we switch the order of the last 6 players, it doesn’t change which team we have determined. So we also have to divide by 6!.
So there are 17! / (11! * 6!) different possible teams.
17! / (11! * 6!) means (17 * 16 * … * 3 * 2 * 1) / (11 * 10 * … * 3 * 2 * 1 * 6 * 5 * 4 * 3 * 2 * 1).
We can cancel the 11 * 10 * … * 3 * 2 * 1 on top and bottom, which leaves us with
(17 * 16 * 15 * 14 * 13 * 12) / (6 * 5 * 4 * 3 * 2 * 1)
Now, 6 * 2 on the bottom cancels with 12 on top, which gives us
(17 * 16 * 15 * 14 * 13) / (5 * 4 * 3)
(Of course, multiplying by 1 doesn’t do anything, so we may as well leave that off.) Now the 5 * 3 cancels with the 15:
(17 * 16 * 14 * 13) / 4
Finally, dividing the 16 by 4 gives
17 * 4 * 14 * 13.
Multiplying the first two together and the last two together (which can be done mentally, or isn’t hard to do with pencil and paper) gives
68 * 182,
which isn’t hard to multiply out using pencil and paper. (The answer is 12,376.)
There are 11 people on a cricket team, so this asks “how many different combinations of 11 elements can be chosen from 17 elements?”
17C11 is another way of writing this.
The expansion of nCm is n! / [m! (n - m)!]
So you have 17! / [11! (6)!]
= 17 x 16 x 15 x 14 x 13 x 12 / (6 x 5 x 4 x 3 x 2)
= 17 x 4 x 14 x 13
= 12376. Easy multiplication without a calculator.
cheers.
The answer is annoying.
Let’s say we have 17 people, each of whose names start with a different letter, from Abraham to Quincy. For the first person, we could select any of 17 people. That leaves 16. For the second, we can select any of 16 people. And so on down until the 11th person, when we can select any of 7 people.
If we were asking these people to stand in a line, the number of possible lines we could get is 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7. This is a huge number: 494,010,316,800.
However, a team is different from a line. Let’s say the first person we picked was Abraham, the second Robert and the third Lydia. Would this be any different from picking Lydia first, Abraham second and Robert third? No.
So we divide that huge number by 11! That is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 = 39,916,800.
And the answer is 12,376.
And when people didn’t have calculators, they usually multiplied by hand. Sometimes a sliderule was useful.
The formula to calculate the combination of 17 players selected 11 at a time is:
n! / r!(n-r)!
n = the number of players to choose from, and r is the number of players you will choose.
Substituting in and doing the subtraction part, we get:
17! / 11! * 6!
To calculate the factorial (17!) you would multiply 17 x 16 x 15….. all the way down to 1. But we can simplify things.
Divide out the 11! and you get:
17 x 16 x 15 x 14 x 13 x 12 divided by 6 x 5 x 4 x 3 x 2 x 1
You could then do some more simplification, or simply do the multiplication and division.
The result is 12, 376 possible combinations.
a lot
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