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In how many different ways can five people be seated in a row if two of them insist on not sitting next to each other?
In how many ways can 3 English, 3 history, and 2 math books be set on a shelf, if the English books are set on the left, history books in the middle, and math books on the right?
72
In how many ways can 3 English, 3 history, and 2 math books be set on a shelf, if they are grouped by subject?
You have 5 math books and 6 history books to put on a shelf with five slots. In how many ways can you put the books on the shelf if the first two slots are to be filled with math books and the next three with history books?
2,400
You have 5 math books and 6 history books to put on a shelf with five slots. In how many ways can you put the books on the shelf if the first two slots are to be filled with the books of one subject and the next three slots are to be filled with the books of the other subject?
Do the following problems using the techniques learned in this section.
In how many different ways can five children hold hands to play "Ring Around the Rosy"?
24
In how many ways can three people be made to sit at a round table?
In how many different ways can six children ride a "Merry Go Around" with six horses?
120
In how many ways can three couples be seated at a round table, so that men and women sit alternately?
In how many ways can six trinkets be arranged on a chain?
120
In how many ways can five keys be put on a key ring?
Find the number of different permutations of the letters of the word MASSACHUSETTS.
64,864,800
Find the number of different permutations of the letters of the word MATHEMATICS.
Seven flags, three red, two white, and two blue, are to be flown on seven poles. How many different arrangements are possible?
210
How many different ways can three pennies, two nickels and five dimes be arranged in a row?
How many four-digit numbers can be made using two 2's and two 3's?
6
How many five-digit numbers can be made using two 6's and three 7's?
If a coin is tossed 5 times, how many different outcomes of 3 heads, and 2 tails are possible?
10
If a coin is tossed 10 times, how many different outcomes of 7 heads, and 3 tails are possible?
If a team plays ten games, how many different outcomes of 6 wins, and 4 losses are possible?
210
If a team plays ten games, how many different ways can the team have a winning season?
Do the following problems using combinations.
How many different 3-people committees can be chosen from ten people?
120
How many different 5-player teams can be chosen from eight players?
In how many ways can a person choose to vote for three out of five candidates on a ballot for a school board election?
10
Compute the following:
$\mathrm{9C2}$
$\mathrm{6C4}$
$\mathrm{8C3}$
$\mathrm{7C4}$
How many 5-card hands can be chosen from a deck of cards?
2,598,960
How many 13-card bridge hands can be chosen from a deck of cards?
There are twelve people at a party. If they all shake hands, how many different hand-shakes are there?
66
In how many ways can a student choose to do four questions out of five on a test?
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