91Ó°ÊÓ

Assume that the probability of the birth of a child of a particular sex is \(50 \% .\) In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) \(3,(b) 8,(c) 12,\) and \((d) 20\).

Find the sum of the finite geometric sequence. $$\sum_{n=0}^{6} 500(1.04)^{n}$$

You are dealt five cards from a standard deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.)

An employer interviews 12 people for four openings at a company. Five of the 12 people are women. All 12 applicants are qualified. In how many ways can the employer fill the four positions when (a) the selection is random and (b) exactly two selections are women?

Write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$\begin{array}{l} a_{1}=0 \\ a_{n}=a_{n-1}+2 n \end{array}$$

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Hexagon

Three points that are not collinear determine three lines. How many lines are determined by nine points, no three of which are collinear?

Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in 42 states, Washington D.C., and the U.S. Virgin Islands. The game is played by drawing five white balls out of a drum of 59 white balls (numbered \(1-59\) ) and one red powerball out of a drum of 35 red balls (numbered 1-35). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers when you win the jackpot by matching all five white balls in order and the red powerball.

Find a formula for the sum of the angles (in degrees) of a regular polygon. Then use mathematical induction to prove this formula for a general \(n\) -sided polygon.