If a team won 3 games and lost 2, how many patterns...?

C

Cherry3451

New member
Joined
Aug 10, 2008
Messages
5
Kathy's softball team has won 3 games and lost 2. How many patterns of records are possible?

I know there is an easier way to do it than to write them all down, but I don't know how. Does someone know the formula or process to solve it? Thanks so much!
 
Hint (1=win, 0=loss):
Code: 
 1 2 3 4 5 : games
 1 1 1 0 0
 1 1 0 0 1
 1 1 0 1 0
......
Kapish?
 
Hello, Cherry3451!

Kathy's softball team has won 3 games and lost 2.
How many patterns of records are possible?
Click to expand...

There is a formula for this problem.

\(\displaystyle \text{There are }n\text{ games and they have won }r\text{ of them.}\)
\(\displaystyle \text{The number of possible patterns of wins/losses is: }\;{n\choose r} \;=\;\frac{n!}{r!(n-r)!}\)
. . I will assume you understand "factorials".

\(\displaystyle \text{The answer is: }\;{5\choose3} \:=\:\frac{5!}{3!2!} \;=\;10\text{ patterns.}\)

. . . . . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We can reason it out ... if we're careful.

\(\displaystyle \text{Call the games: }\:A, B, C, D, E\)

\(\displaystyle \text{There are two losses. }\:\text{Where can they appear?}\)

\(\displaystyle \text{For the first loss, there is a choice of 5 games.}\)
\(\displaystyle \text{For the second loss, there is a choice of 4 games.}\)
. . \(\displaystyle \text{It seems that there are: }5\times 4 \:=\:20\text{ ways.}\)

\(\displaystyle \text{But this list includes: "lose games A and B" and "lose games B and A".}\)
. . \(\displaystyle \text{Since these are the same pattern, there is much duplication in our list.}\)
. . \(\displaystyle \text{In fact, there are }twice\text{ as many patterns in our list than there should be.}\)

\(\displaystyle \text{Therefore, the number of patterns is: }\:\frac{20}{2} \;=\;10\)

 
You must log in or register to reply here.
Top