Question 1

Use the binomial theorem to expand and simplify.

Accepted Answer

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

Question 2

Evaluate the expression. 6!0!

Accepted Answer

The expression 6!0! is equal to 6! * 0!. The factorial of 6 (6!) is 720, and the factorial of 0 (0!) is 1. Therefore, 6!0! = 720 * 1 = 720.

Question 3

If ten 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 number of different ways to decide first, second, and third place among ten teams, without allowing ties, can be calculated using permutations. This is because the order in which the teams are placed matters. The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items (in this case, teams) and r is the number of items to be arranged. Here, n = 10 (teams) and r = 3 (places to be decided). Thus, the calculation is 10P3 = 10! / (10 - 3)! = 10! / 7! = (10 × 9 × 8 × 7!) / 7! = 720.

Question 4

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

Accepted Answer

Each pair of points determines a unique line, and with 9 points, we can form $\binom{9}{2} = 36$ unique pairs, hence 36 lines.

Question 5

Express the sum in terms of n.  (Hint: Use the theorem on sums to write the sum as  . Next, use the following formulas  and the theorem on the sum of a constant.)

Accepted Answer

The answer of Express the sum in terms of n....

Question 6

In a round-robin tennis tournament, every player meets every other player exactly once. How many players can participate in a tournament of 78 matches?

Accepted Answer

In a round-robin tournament with n players, each player plays against every other player exactly once. This results in a total of n(n-1)/2 matches. Setting this equal to 78 gives us the equation n(n-1)/2 = 78. Solving for n, we find that n = 13.

Question 7

Use the principle of mathematical induction to find the factor of the expression.  Hint:

Accepted Answer

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

Question 8

Use the binomial theorem to expand and simplify.

Accepted Answer

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

Question 9

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 selection 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. For this problem, n = 45 and r = 6, so the calculation is C(45, 6) = 45! / [6!(45 - 6)!] = 8,145,060.

Question 10

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 6th term of the sequence.

Accepted Answer

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

Question 11

Find the number of terms in the arithmetic sequence with the given conditions.  ,

Accepted Answer

The answer of Find the number of terms in the...

Question 12

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

Accepted Answer

The answer of A girl has three skirts, ten blouses...

Question 13

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...

Question 14

Find the specified term of the arithmetic sequence that has the two given terms.

Accepted Answer

The answer of Find the specified term of the arithmetic...

Question 15

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 16

Use the binomial theorem to expand and simplify.

Accepted Answer

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

Question 17

Evaluate the expression.

Accepted Answer

The answer of Evaluate the expression. 
...

Question 18

Find the number.

Accepted Answer

The answer of Find the number. 
...

Question 19

Find the sum.

Accepted Answer

The answer of Find the sum. 
...

Question 20

Find the fourth term of the recursively defined infinite sequence.

Accepted Answer

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

Quiz 10: Sequences, Series and Probability

Use the binomial theorem to expand and simplify.

Evaluate the expression. 6!0!

If ten 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.

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

Express the sum in terms of n. (Hint: Use the theorem on sums to write the sum as . Next, use the following formulas and the theorem on the sum of a constant.)

In a round-robin tennis tournament, every player meets every other player exactly once. How many players can participate in a tournament of 78 matches?

Use the principle of mathematical induction to find the factor of the expression. Hint:

Use the binomial theorem to expand and simplify.

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.

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 6th term of the sequence.

Find the number of terms in the arithmetic sequence with the given conditions. ,

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

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

Find the specified term of the arithmetic sequence that has the two given terms.

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

Use the binomial theorem to expand and simplify.

Evaluate the expression.

Find the number.

Find the sum.

Find the fourth term of the recursively defined infinite sequence.