Question 1

Solve the problem.
-How many 4-letter codes can be formed using the letters A, B, C, D, E, and F? No letter can be used more than once.

Accepted Answer

Since no letter can be used more than once and there are 6 letters to choose from, the number of 4-letter codes can be calculated using permutations: 6P4 = 6! / (6-4)! = 6*5*4*3 = 360.

Question 2

Find a Particular Term in a Binomial Expansion
-$$( 3 x + 2 y ) ^ { 8 }$$

Accepted Answer

The correct answer is A because it is the only choice that matches the given binomial expansion.

Question 3

Find the term indicated in the expansion.
-$( x - 2 y ) ^ { 11 } ; 8$ th term

Accepted Answer

The 8th term in the expansion of $(x - 2y)^{11}$ can be found using the binomial theorem formula for the $n$th term: $T_{k+1} = \binom{n}{k} x^{n-k} y^k$, where $n = 11$ and $k = 8 - 1 = 7$. Thus, $T_8 = \binom{11}{7} x^{11-7} (-2y)^7 = \binom{11}{7} x^4 (-2)^7 y^7$. Calculating $\binom{11}{7} = 330$ and $(-2)^7 = -128$, the term becomes $-42,240 x^4 y^7$.

Question 4

Use the Fundamental Counting Principle
Solve the problem.
-A student must choose 1 of 5 science electives, 1 of 6 social studies electives, and 1 of 9 language electives. How many possible course selections are there?

Accepted Answer

The Fundamental Counting Principle states that if there are n ways to do one thing, and m ways to do another, then there are n*m ways to do both. Here, the student has 5 choices for science, 6 for social studies, and 9 for language, so the total number of course selections is 5*6*9 = 270.

Question 5

Find the term indicated in the expansion.
-$( 2 x + 4 ) 5$; 5 th term

Accepted Answer

The binomial expansion of $(2x + 4)^5$ is given by the binomial theorem. The 5th term corresponds to $T_5 = \binom{5}{4} (2x)^1 (4)^4 = 5 \cdot 2x \cdot 256 = 2560x$.

Question 6

Find the term indicated in the expansion.
-$( 3 \mathrm { x } + 2 ) ^ { 5 }$; 5 th term

Accepted Answer

The 5th term in the expansion of $(3x + 2)^5$ can be found using the binomial theorem formula for the $n$th term: $T_{k+1} = \binom{n}{k} a^{n-k} b^k$, where $n=5$ and $k=4$ for the 5th term. Thus, $T_5 = \binom{5}{4} (3x)^{1} (2)^{4} = 5 \cdot 3x \cdot 16 = 240x$.

Question 7

Find the term indicated in the expansion.
-$( 2 x - 3 y ) ^ { 11 ; }$ 4th term

Accepted Answer

The 4th term in the expansion of $(2x - 3y)^{11}$ can be found using the binomial theorem formula for the $k$th term: $_nC_k \cdot a^{n-k} \cdot b^k$, where $n$ is the power of the binomial, $a$ and $b$ are the terms of the binomial, and $k$ is the term number minus 1. For the 4th term, $k = 3$, so we calculate $_{11}C_3 \cdot (2x)^{11-3} \cdot (-3y)^3$. This simplifies to $165 \cdot 2^8 x^8 \cdot (-27)y^3$, which equals $-1,140,480 x^8 y^3$.

Question 8

Use the Permutations Formula
Use the formula for
 $${ } _ { n } P _ { r }$$ to evaluate the expression.
-$$10 \mathrm { P } _ { 4 }$$

Accepted Answer

The formula for permutations is ${ } _ { n } P _ { r } = \frac{n!}{(n-r)!}$. For $10P4$, it is $\frac{10!}{(10-4)!} = \frac{10!}{6!}$. Calculating this gives $10 \times 9 \times 8 \times 7 = 5040$.

Question 9

Find a Particular Term in a Binomial Expansion
-$$\left( x ^ { 2 } + 3 \right) ^ { 9 }$$

Accepted Answer

The correct term can be found using the binomial theorem, which states that the $k$th term in the expansion of $(a + b)^n$ is given by $\binom{n}{k}a^{n-k}b^k$. For the term $x^{16}$, $k = 1$ since we are decreasing the power of $x^2$ by one step (from $x^{18}$ to $x^{16}$), so the coefficient is $\binom{9}{1}3^1 = 9 \times 3 = 27$. The term $x^{14}$ corresponds to $k = 2$, so its coefficient is $\binom{9}{2}3^2 = 36 \times 9 = 324$. Therefore, the correct expansion includes these terms as part of the sequence, matching option A.

Question 10

Use the Fundamental Counting Principle
Solve the problem.
-In how many ways can 5 volunteers be assigned to 5 booths for a charity bazaar?

Accepted Answer

The Fundamental Counting Principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.

In this problem, there are 5 volunteers and 5 booths. For the first booth, there are 5 different volunteers who can be assigned to it. For the second booth, there are only 4 volunteers left to choose from, since one has already been assigned. This continues until all 5 booths have a volunteer assigned to them. Therefore, the total number of ways to assign the volunteers to the booths is:
5 x 4 x 3 x 2 x 1 = 120

So, the answer is choice A) 120 ways.

Question 11

Use the Fundamental Counting Principle
Solve the problem.
-A restaurant offers a choice of 3 salads, 10 main courses, and 4 desserts. How many possible choices for a meal are there (including single items)?

Accepted Answer

To find the total number of possible meal choices, we multiply the number of choices for each course: 
3 salads × 10 main courses × 4 desserts = 120 possible combinations. However, we need to remember to include the option of just choosing a single item from any of the courses, so we add those possibilities: 
3 salads + 10 main courses + 4 desserts = 17 single item choices. 
Therefore, the total number of possible meal choices is 120 + 17 = 219.

Question 12

Solve the problem.
-How many 3-letter codes can be formed using the letters A, B, C, D, and E? No letter can be used more than once.

Accepted Answer

Since no letter can be used more than once and we need to form 3-letter codes from 5 different letters, we use permutations. The number of ways to arrange 5 items in groups of 3 is 5P3 = 5! / (5-3)! = 5*4*3 = 60.

Question 13

Use the Fundamental Counting Principle
Solve the problem.
-There are 5 roads leading from Bluffton to Hardeeville, 7 roads leading from Hardeeville to Savannah, and 3 roads leading from Savannah to Macon. How many ways are there to get from Bluffton to Macon?

Accepted Answer

To find the total number of ways to get from Bluffton to Macon, multiply the number of roads for each segment of the trip: 5 roads (Bluffton to Hardeeville) * 7 roads (Hardeeville to Savannah) * 3 roads (Savannah to Macon) = 105 ways.

Question 14

Find the term indicated in the expansion.
-$( x + 3 y ) ^ { 9 } ;$ 4th term

Accepted Answer

The 4th term in the expansion of $(x + 3y)^9$ can be found using the binomial theorem formula for the $k$th term: $_{n}C_{k-1} \cdot x^{n-(k-1)} \cdot y^{k-1}$, where $n=9$ and $k=4$. Thus, the 4th term is $_{9}C_{3} \cdot x^{9-3} \cdot (3y)^{3} = 84 \cdot x^6 \cdot 27y^3 = 2268x^6y^3$.

Question 15

Use the Fundamental Counting Principle
Solve the problem.
-In how many ways can 7 players be assigned to 7 positions on a baseball team, assuming that any player can play any position?

Accepted Answer

The Fundamental Counting Principle states that if there are m ways to do one thing and n ways to do another, then there are m x n ways to do both. In this problem, there are 7 players who can be assigned to any of the 7 positions, so there are 7 options for the first player, 6 options for the second player (since one position has been filled), 5 options for the third player, and so on until there is only one player left to be assigned to the remaining position. This can be represented as 7 x 6 x 5 x 4 x 3 x 2 x 1, which simplifies to 5040 ways. Therefore, the best choice is A) 5040 ways.

Question 16

Use the Fundamental Counting Principle
Solve the problem.
-A restaurant offers a choice of 5 salads, 9 main courses, and 4 desserts. How many possible 3-course meals are there?

Accepted Answer

The number of possible 3-course meals is found by multiplying the number of choices for each course: 5 salads * 9 main courses * 4 desserts = 180 possible meals.

Question 17

Find a Particular Term in a Binomial Expansion
-$( x - 3 ) ^ { 17 }$

Accepted Answer

The correct expansion for the given binomial expression $(x - 3)^{17}$ starts with $x^{17}$, and the coefficients of the subsequent terms can be calculated using the binomial theorem. The coefficient of $x^{16}$ is $-51$ because it is $17$ (from $17C1$) multiplied by $-3$ (the second term of the binomial raised to the first power). Similarly, the coefficient of $x^{15}$ is $1224$, calculated from $17C2$ times $(-3)^2$, which gives a positive value. Thus, option A correctly represents the first three terms of the expansion.

Question 18

Use the Fundamental Counting Principle
Solve the problem.
-Lisa has 5 skirts, 10 blouses, and 4 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

Accepted Answer

Using the Fundamental Counting Principle, multiply the number of choices for each piece of clothing: 5 skirts * 10 blouses * 4 jackets = 200 possible outfits.

Question 19

Solve the problem.
-${ } _ { 8 } \mathrm { P } _ { 0 }$

Accepted Answer

The permutation formula ${ }_n\mathrm{P}_r = \frac{n!}{(n-r)!}$ simplifies to 1 when $r = 0$, because $n! / n! = 1$.

Question 20

Find the term indicated in the expansion.
-$\left( x ^ { 4 } + y ^ { 3 } \right) ^ { 9 } : 3$ rd term

Accepted Answer

The third term in the expansion of $\left( x ^ { 4 } + y ^ { 3 } \right) ^ { 9 } $ is given by the formula:$$T_3 = \binom{9}{2} (x^4)^7 (y^3)^2 = 36 x^{28} y^6$$

Quiz 8: Sequences, Induction, and Probability

Solve the problem. -How many 4-letter codes can be formed using the letters A, B, C, D, E, and F? No letter can be used more than once.

Find a Particular Term in a Binomial Expansion - $( 3 x + 2 y ) ^ { 8 }$

Find the term indicated in the expansion. - $( x - 2 y ) ^ { 11 } ; 8$ th term

Use the Fundamental Counting Principle Solve the problem. -A student must choose 1 of 5 science electives, 1 of 6 social studies electives, and 1 of 9 language electives. How many possible course selections are there?

Find the term indicated in the expansion. - $( 2 x + 4 ) 5$ ; 5 th term

Find the term indicated in the expansion. - $( 3 \mathrm { x } + 2 ) ^ { 5 }$ ; 5 th term

Find the term indicated in the expansion. - $( 2 x - 3 y ) ^ { 11 ; }$ 4th term

Use the Permutations Formula Use the formula for ${ } _ { n } P _ { r }$ to evaluate the expression. - $10 \mathrm { P } _ { 4 }$

Find a Particular Term in a Binomial Expansion - $\left( x ^ { 2 } + 3 \right) ^ { 9 }$

Use the Fundamental Counting Principle Solve the problem. -In how many ways can 5 volunteers be assigned to 5 booths for a charity bazaar?

Use the Fundamental Counting Principle Solve the problem. -A restaurant offers a choice of 3 salads, 10 main courses, and 4 desserts. How many possible choices for a meal are there (including single items) ?

Solve the problem. -How many 3-letter codes can be formed using the letters A, B, C, D, and E? No letter can be used more than once.

Use the Fundamental Counting Principle Solve the problem. -There are 5 roads leading from Bluffton to Hardeeville, 7 roads leading from Hardeeville to Savannah, and 3 roads leading from Savannah to Macon. How many ways are there to get from Bluffton to Macon?

Find the term indicated in the expansion. - $( x + 3 y ) ^ { 9 } ;$ 4th term

Use the Fundamental Counting Principle Solve the problem. -In how many ways can 7 players be assigned to 7 positions on a baseball team, assuming that any player can play any position?

Use the Fundamental Counting Principle Solve the problem. -A restaurant offers a choice of 5 salads, 9 main courses, and 4 desserts. How many possible 3-course meals are there?

Find a Particular Term in a Binomial Expansion - $( x - 3 ) ^ { 17 }$

Use the Fundamental Counting Principle Solve the problem. -Lisa has 5 skirts, 10 blouses, and 4 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

Solve the problem. - ${ } _ { 8 } \mathrm { P } _ { 0 }$

Find the term indicated in the expansion. - $\left( x ^ { 4 } + y ^ { 3 } \right) ^ { 9 } : 3$ rd term

Quiz 8: Sequences, Induction, and Probability

Solve the problem. -How many 4-letter codes can be formed using the letters A, B, C, D, E, and F? No letter can be used more than once.

Find a Particular Term in a Binomial Expansion - (3x+2y) 8( 3 x + 2 y ) ^ { 8 }(3x+2y) 8

Find the term indicated in the expansion. - (x−2y) 11;8( x - 2 y ) ^ { 11 } ; 8(x−2y) 11;8 th term

Use the Fundamental Counting Principle Solve the problem. -A student must choose 1 of 5 science electives, 1 of 6 social studies electives, and 1 of 9 language electives. How many possible course selections are there?

Find the term indicated in the expansion. - (2x+4) 5( 2 x + 4 ) 5(2x+4) 5 ; 5 th term

Find the term indicated in the expansion. - (3x+2) 5( 3 \mathrm { x } + 2 ) ^ { 5 }(3x+2) 5 ; 5 th term

Find the term indicated in the expansion. - (2x−3y) 11;( 2 x - 3 y ) ^ { 11 ; }(2x−3y) 11; 4th term

Use the Permutations Formula Use the formula for nPr{ } _ { n } P _ { r }n​Pr​ to evaluate the expression. - 10P410 \mathrm { P } _ { 4 }10P4​

Find a Particular Term in a Binomial Expansion - (x2+3) 9\left( x ^ { 2 } + 3 \right) ^ { 9 }(x2+3) 9