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From a pool of 7 secretaries, 3 are selected to be assigned to 3 managers, 1 secretary to each manager. In how many ways can this be done?

A population of fruit flies is growing in such a way that each generation is 1.25 times as large as the last generation. Suppose there were 200 insects in the first generation. How many would there be in the fifth generation?

In a game of musical chairs, 13 children will sit in 12 chairs. (1 will be left out.) How many seating arrangements are possible?

Each year a machine loses \(20 \%\) of the value it had at the beginning of the year. Find the value of the machine at the end of 6 yr if it cost 100,000 dollars new.

In a club with 8 women and 11 men members, how many 5 -member committees can be chosen that have the following? (a) all women (b) all men (c) 3 women and 2 men (d) no more than 3 men

The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

To win the jackpot in a lottery game, a person must pick 4 numbers from 0 to 9 in the correct order. If a number can be repeated, how many ways are there to play the game?

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{5}(5 i+3)$$

(Modeling) Children's Growth Pattern The normal growth pattern for children aged \(3-11\) follows that of an arithmetic sequence. An increase in height of about \(6 \mathrm{cm}\) per year is expected. Thus, 6 would be the common difference of the sequence. For example, a child who measures \(96 \mathrm{cm}\) at age 3 would have his expected height in subsequent years represented by the sequence \(102,108,114,120\) \(126,132,138,144 .\) Each term differs from the adjacent terms by the common difference, 6 (a) If a child measures \(98.2 \mathrm{cm}\) at age 3 and \(109.8 \mathrm{cm}\) at age \(5,\) what would be the common difference of the arithmetic sequence describing his yearly height? (b) What would we expect his height to be at age \(8 ?\) (IMAGE CANT COPY)

Prove statement for positive integers \(n\) and \(r,\) with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$\left(\begin{array}{l}n \\\0\end{array}\right)=1$$