i need help with this combination/probability thingy :(

imverystupid · #1 $Old$ November 13th, 2009, 07:06 PM

i know how to solve these questions, but i'm supposed to put them into categories but i don't know what kind of categories to put them into

the teacher also says to write what these problems have in common and why you multiply to get the answers but i am very stupid so i don't know.

1. A deli has five types of meat, two types of cheese, and three types of bread. How many different sandwiches consisting of one type of meat, one type of cheese, and one type of bread, does the deli serve?

2. In a six-team division of a baseball league, each team plays each other 9 times. How many games will be played in the division?

3. A standard 52-card deck has four suits (hearts, clubs, diamonds, and spades) with 13 cards in each suit. How many five-card hands consist of four diamonds and one spade?

4. John has five shirts, three ties, and seven suits. How many possible outfits consisting of one shirt, one tie, and one suit does he have?

5. A locker combination system uses three digits from 0 to 9. How many different three-digit combinations with no digit repeated are possible?

Prove It · #2 $Old$ November 13th, 2009, 07:09 PM

Quote:

Originally Posted by imverystupid $View Post$

i know how to solve these questions, but i'm supposed to put them into categories but i don't know what kind of categories to put them into

the teacher also says to write what these problems have in common and why you multiply to get the answers but i am very stupid so i don't know.

1. A deli has five types of meat, two types of cheese, and three types of bread. How many different sandwiches consisting of one type of meat, one type of cheese, and one type of bread, does the deli serve?

2. In a six-team division of a baseball league, each team plays each other 9 times. How many games will be played in the division?

3. A standard 52-card deck has four suits (hearts, clubs, diamonds, and spades) with 13 cards in each suit. How many five-card hands consist of four diamonds and one spade?

4. John has five shirts, three ties, and seven suits. How many possible outfits consisting of one shirt, one tie, and one suit does he have?

5. A locker combination system uses three digits from 0 to 9. How many different three-digit combinations with no digit repeated are possible?

Use tree diagrams to determine all the possible combinations.

Soroban · #3 $Old$ November 14th, 2009, 05:48 AM

Hello, imverystupid!

Quote:

i'm supposed to put these into categories,
but i don't know what kind of categories to put them into.
I don't know the categories either . . . can't help you.

The teacher also says to write what these problems have in common
They're "counting problems"?

and why you multiply to get the answers.
Because of the "Fundamental Theorem of Counting"?

Quote:

1. A deli has 5 types of meat, 2 types of cheese, and 3 types of bread.
How many different sandwiches consist of 1 meat, 1 cheese, and 1 bread,?

This is a "fundamental" problem.

$5\times 2 \times 3 \:=\:30$ possible sandwiches.

Quote:

2. In a six-team division of a baseball league, each team plays each other 9 times.
How many games will be played in the division?

This is "combinations".

The 6 teams can be paired off in: . $_6C_2 \:=\:{6\choose2} \:=\:\frac{6!}{2!\,4!} \:=\:15$ ways.

This pairing will be repeated 9 times.

There will be: . $9 \times 15 \:=\:135$ games.

Quote:

3. A standard 52-card deck has four suits (hearts, clubs, diamonds, and spades)
with 13 cards in each suit. How many five-card hands consist of four Diamonds and one Spade?

This is "combinations".

There are: . $_{13}C_4 \;=\;\frac{13!}{4!\,9!} \;=\;{13\choose4} \;=\;715$ ways to get 4 $\diamondsuit s$
There are: . $_{13}C_1 \;=\;{13\choose1} \;=\;13$ ways to get one $\spadesuit$

Therefore, there are: . $715 \times 13 \:=\:9295$ hands with 4 $\diamondsuit \text{s and 1 }\spadesuit.$

Quote:

4. John has five shirts, three ties, and seven suits.
How many possible outfits have 1 shirt, 1 tie, and 1 suit?

Another "fundamental" problem.

He has: . $5 \times 3 \times 7 \:=\:105$ possible outfits.

Quote:

5. A locker combination system uses three digits from 0 to 9.
How many different three-digit combinations with no digit repeated are possible?

Yet another "fundamental" problem.

The 1st digit has 10 choices.
The 2nd digit has 9 choices.
The 3rd digit has 8 choices.

Therefore, there are: . $10 \times 9 \times 8 \:=\:720$ possible combinations.