3 in how many ways can 6 students be seated in a row of 6...

3). In how many ways can 6 students be seated in a row of 6 chairs if Jack insists on sitting in the first chair? 4). A president, a treasurer, and a secretary are to be chosen from a committee with forty members. In how many ways could the three officers can be chosen? 5). In how many ways can 7 books be chosen from a group of nine? 9). Suppose that a department contain 13 men and 15 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

Transcript

00:01 Our first question here asks, and it's question three, in how many ways can six students be seated in a row of six chairs if jack insists on sitting in the first chair? so we have six people.

00:14 One of those people is jack.

00:19 The other five don't care where they sit, but we have one, two, three, four, five, six spots.

00:29 Jack is going to sit here, so there's only one option for one spot because he insists on sitting in the first chair.

00:36 The other spots have the choice of the rest of the five people.

00:42 So spot two has the choice of five people and then four, three, two, one.

00:48 The remaining people are going to sit in the other spots and we multiply these numbers together.

00:56 One times five is five.

01:00 And then we take five times four is 20.

01:03 And then we take that number, 20 times 3 is 60, 60 times 2 is 120, and 120 times 1 is 120.

01:15 So there are 120 ways in which these people could sit with jack insisting on sitting in that first spot.

01:28 Question 4 asks us, says a president, a treasurer, and a secretary are to be chosen from a committee of 40 people.

01:38 In how many ways could the three officers be chosen? so we have the president, we have a treasurer, and we have a secretary.

01:53 And there's a total of 40 people to choose from.

01:56 So if we choose the president first, there's 40 choices for president.

02:01 Once they're chosen, now there's 39 remaining to be chosen for treasurer.

02:05 And once they're chosen, there's 38 people left to be chosen for secretary.

02:10 And we multiply these numbers together.

02:14 So we get here, 40 times 39 times 38.

02:22 40 times 39 times 38 gives us 59, 280 ways in which people could be chosen to fill those three spots.

02:33 That's how many combinations we have.

02:41 The next question, which is question five, in how many ways can seven books be chosen from a group of nine? so we have book one, book two, book three, book four, book five, book six, and book seven that we're choosing...

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