Permutations/Combinatorics (students taking courses)

[lala pre calc land]

If someone could please help on the following problems:

1) All students must take at least 1 of the 3 following courses: bio, chem, and physics. Out of 297 students, the following results were gathered
132 intend to take bio and chem
107 intend to take chem and physics
88 intend to take bio and physics
43 intend to take only bio
55 intend to take only chem
38 intend to take only physics

How many students intend to take all 3 courses?

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2) In the World Series, 2 teams, A and B, play each other until 1 team has won 4 games. For example, the "word" ABBAAA represents a 6 game series in which team A wins games 1, 4, 5, and 6
Explain why the number of different 6 game series won by team A is:
5! divided by (3! * 2!)

Show that there are 70 different sequences of games possible

Thanks!

 

[pka]

Re: Permutations/Combinatorics

1) All students must take at least 1 of the 3 following courses: bio, chem, and physics. Out of 297 students, the following results were gathered
132 intend to take bio and chem
107 intend to take chem and physics
88 intend to take bio and physics
43 intend to take only bio
55 intend to take only chem
38 intend to take only physics
How many students intend to take all 3 courses?

In the picture you are looking for x.
From “43 intend to take only bio” you know that v=43.
From “132 intend to take bio and chem” you know that x+t=132.
Now you work it out.

2) In the World Series, 2 teams, A and B, play each other until 1 team has won 4 games. For example, the "word" ABBAAA represents a 6 game series in which team A wins games 1, 4, 5, and 6
Explain why the number of different 6 game series won by team A is: 5! divided by (3! * 2!)

That is the number of ways to rearrange the string “BBAAA” or B wins two and A wins three then A wins the sixth game played.

 

[soroban]

Re: Permutations/Combinatorics

Hello, lala pre calc land!

2) In the World Series, 2 teams, A and B, play each other until a team has won 4 games.
For example, the "word" ABBAAA represents a 6 game series
in which team A wins games 1, 4, 5, and 6.

(a) Explain why the number of different 6 game series won by team A is: \(\displaystyle \L\:\frac{5!}{3!2!}\)

(b) Show that there are 70 different sequences of games possible.

(a) To have a 6-game series, A's fourth win must be the sixth game.
Of the first five games: A won three games and B won two games.
And this can happen in \(\displaystyle \L\,\frac{5!}{3!2!}\,=\,10\) ways.


(b) Consider the way in which \(\displaystyle A\) can win the World Series.

\(\displaystyle A\) wins a 4-game series: \(\displaystyle \L\,1\) way.

\(\displaystyle A\) wins a 5-game series
Of the first 4 games, \(\displaystyle A\) has won 3 and \(\displaystyle B\) has won one game.
There are: \(\displaystyle \L\,\frac{4!}{3!1!}\,=\,4\) ways.

\(\displaystyle A\) win a 6-game series.
We already know this happens in \(\displaystyle \L\,\frac{5!}{3!2!}\,=\,10\) ways.

\(\displaystyle A\) win a 7-game series.
Of the first 6 games, \(\displaystyle A\) has won 3 and \(\displaystyle B\) has won 3 games.
There are: \(\displaystyle \L\,\frac{6!}{3!3!} \,=\,20\) ways.

Hence, \(\displaystyle A\) can win the series in: \(\displaystyle \L\,1\,+\,4\,+\,10\,+\,20\:=\:35\) ways.
. . And \(\displaystyle B\) can also win the series in \(\displaystyle \L\,35\) ways.

Therefore, there are: \(\displaystyle \L\,35\,+\,35\:=\:70\) different possible game sequences.