Question 1

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 45. Find the total number of selections possible.

Accepted Answer

The total number of selections possible is calculated using the combination formula since the order of the numbers selected does not matter. The formula for combinations is C(n, r) = n! / [r!(n - r)!], where n is the total number of items to choose from, r is the number of items to choose, and "!" denotes factorial. Here, n = 45 and r = 6. Therefore, the total number of selections possible is C(45, 6) = 45! / [6!(45 - 6)!] = 8,145,060.

Question 2

Use the binomial theorem to expand and simplify.

Accepted Answer

The answer of Use the binomial theorem to expand and...

Question 3

Find the sum.

Accepted Answer

The answer of Find the sum. 
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Question 4

Consider any six points such that no three are collinear. How many lines are determined?

Accepted Answer

A line is determined by any two points. With six points and no three being collinear, we can calculate the number of lines using the combination formula for choosing 2 points out of 6, which is 6C2 = (6*5)/(2*1) = 15.

Question 5

A company is to distribute $28,000 in bonuses to its top five salespeople. The fifth salesperson on the list will receive $4,400, and the difference in bonus money between successively ranked salespeople is to be constant. Find the bonus for each salesperson.

Accepted Answer

The total bonus amount is $28,000, and the difference in bonuses is constant. Starting from the fifth salesperson with $4,400 and working backwards with a constant difference, option E correctly sums to $28,000 and maintains a constant difference of $600 between each salesperson.

Question 6

Bode's sequence, defined by  can be used to approximate distances of planets from the sun. These distances are measured in astronomical units, with 1 AU = 93,000,000 mi. For example, the third term corresponds to earth and the fifth term to the minor planet Ceres. Find the 5th term of the sequence.

Accepted Answer

The answer of Bode's sequence, defined by  can be...

Question 7

Use the binomial theorem to expand and simplify.

Accepted Answer

The answer of Use the binomial theorem to expand and...

Question 8

Find the fourth partial sum for the given sequence.

Accepted Answer

The answer of Find the fourth partial sum for the...

Question 9

A rubber ball is dropped from a height of 45 feet. If it rebounds approximately two-thirds the distance after each fall, use an infinite geometric series to approximate the total distance the ball travels.

Accepted Answer

The answer of A rubber ball is dropped from a...

Question 10

A single card is drawn from a deck. Find the probability that the card is either red or a two.

Accepted Answer

The answer of A single card is drawn from a...

Question 11

Find the third term of the sequence:  is the number of decimal places in  .

Accepted Answer

The answer of Find the third term of the sequence:...

Question 12

A girl has three skirts, six blouses and two sweaters. How many different skirt-blouse-sweater combinations can she wear?

Accepted Answer

The number of combinations is found by multiplying the number of options for each category together: 3 skirts * 6 blouses * 2 sweaters = 36 combinations.

Question 13

Find the fifth term of the geometric sequence whose first two terms are 3 and 6.

Accepted Answer

The common ratio (r) is 6/3 = 2. The nth term of a geometric sequence is given by ar^(n-1), where a is the first term. So, the 5th term is 3*(2)^(5-1) = 3*16 = 48.

Question 14

Find the third term of the recursively defined infinite sequence.

Accepted Answer

The answer of Find the third term of the recursively...

Question 15

Find the sum of the arithmetic sequence  that satisfies the stated conditions.

Accepted Answer

The answer of Find the sum of the arithmetic sequence...

Question 16

Use the principle of mathematical induction to find the equivalent expression for every positive integer n.

Accepted Answer

The answer of Use the principle of mathematical induction to...

Question 17

Find the number.

Accepted Answer

The answer of Find the number. 
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Question 18

If eleven basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Accepted Answer

The answer of If eleven basketball teams are in a...

Question 19

In a round-robin tennis tournament, every player meets every other player exactly once. How many players can participate in a tournament of 66 matches?

Accepted Answer

The answer of In a round-robin tennis tournament, every player...

Question 20

Without expanding completely, find the term in the expansion of the expression that contains  .

Accepted Answer

The answer of Without expanding completely, find the term in...

Quiz 10: Sequences, Series and Probability

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 45. Find the total number of selections possible.

Use the binomial theorem to expand and simplify.

Find the sum.

Consider any six points such that no three are collinear. How many lines are determined?

A company is to distribute $28,000 in bonuses to its top five salespeople. The fifth salesperson on the list will receive $4,400, and the difference in bonus money between successively ranked salespeople is to be constant. Find the bonus for each salesperson.

Bode's sequence, defined by can be used to approximate distances of planets from the sun. These distances are measured in astronomical units, with 1 AU = 93,000,000 mi. For example, the third term corresponds to earth and the fifth term to the minor planet Ceres. Find the 5th term of the sequence.

Use the binomial theorem to expand and simplify.

Find the fourth partial sum for the given sequence.

A rubber ball is dropped from a height of 45 feet. If it rebounds approximately two-thirds the distance after each fall, use an infinite geometric series to approximate the total distance the ball travels.

A single card is drawn from a deck. Find the probability that the card is either red or a two.

Find the third term of the sequence: is the number of decimal places in .

A girl has three skirts, six blouses and two sweaters. How many different skirt-blouse-sweater combinations can she wear?

Find the fifth term of the geometric sequence whose first two terms are 3 and 6.

Find the third term of the recursively defined infinite sequence.

Find the sum of the arithmetic sequence that satisfies the stated conditions.

Use the principle of mathematical induction to find the equivalent expression for every positive integer n.

Find the number.

If eleven basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

In a round-robin tennis tournament, every player meets every other player exactly once. How many players can participate in a tournament of 66 matches?

Without expanding completely, find the term in the expansion of the expression that contains .