Connexus Math 7B unit 2 lesson 6 Permutations Quick Check
1. how many ways can 6 books be arranged on a shelf?
a. 21
b. 120
c. 360
d. 720
2. There are 8 girls competing in an ice-skating competition. In how many ways can they finish in first, second, and third place?
a. 336
b. 1,680
c. 2,016
d. 40,320
3. How many ways can a president and vice president be selected from a class of 12?
a. 23
b. 72
c. 132
d. 1,320
4. How many ways can 7 students line up for a class picture?
a. 5,040
b. 210
c. 28
d.7
Please help idk how to find these answers
1. how many ways can 6 books be arranged on a shelf?
d. 720
(To find the number of ways to arrange the books, you would use the formula for permutations which is n! (n factorial). In this case, since there are 6 books, you would calculate 6! (6 x 5 x 4 x 3 x 2 x 1) which equals 720.)
2. There are 8 girls competing in an ice-skating competition. In how many ways can they finish in first, second, and third place?
a. 336
(For this problem, you would use the permutation formula for picking items from a group: n!/(n-r)! where n is the total number of items and r is the number of items being picked. In this case, n=8 and r=3, so you would calculate 8!/(8-3)! or 8!/(5!) which is equivalent to 8 x 7 x 6 = 336.)
3. How many ways can a president and vice president be selected from a class of 12?
c. 132
(For this problem, n=12 and r=2, so you would calculate 12!/(12-2)! or 12!/(10!) which is equivalent to 12 x 11 = 132.)
4. How many ways can 7 students line up for a class picture?
a. 5,040
(Similar to question 1, you would use the formula for permutations which is n!. In this case, there are 7 students so you would calculate 7! (7 x 6 x 5 x 4 x 3 x 2 x 1) which equals 5,040.)
For questions 1-2, simplify each expression.
1. 9!
A.45
B.362,880
C.265,456
D.125,936
2.25P3
A.12,755
B.14,290
C.13,800
D.18.250
3. Which expression would you use to figure out the number of ways you can arrange the letters in the word equation?
A. 8!
B. 8P4
C. 4P8
D.4!
4. Five friends are having their picture taken. How many ways can the photographer arrange the friends in a row?
A.150
B.100
C.120
D.80
5.How would you apply the fundamental counting principle to a bike combination with 5 numbers?
A. 10+10+10+10+10
B. 10 x 5
C. 10 x 10 x 10 x 10 x 10
D. None of these
1. B. 362,880
2. C. 13,800
3. A. 8!
4. C. 120
5. C. 10 x 10 x 10 x 10 x 10
Can u tell me how you got the answers?
Yes, sure! Here are the explanations for each question:
1. 9!
To simplify 9!, we just need to evaluate the expression. 9! means 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, which equals 362,880. Therefore, the answer is B. 362,880.
2. 25P3
To simplify 25P3, we need to use the permutation formula: nPr = n! / (n - r)!. In this case, n = 25 and r = 3. So, 25P3 = 25! / (25 - 3)! = 25! / 22! = 25 x 24 x 23 = 13,800. Therefore, the answer is C. 13,800.
3. Which expression would you use to figure out the number of ways you can arrange the letters in the word equation?
To find the number of ways to arrange the letters in the word equation, we need to use the permutation formula: nPr = n! / (n - r)!. In this case, there are 8 letters in the word equation, so we need to find the number of permutations of 8 things taken 8 at a time. Therefore, the answer is A. 8!.
4. Five friends are having their picture taken. How many ways can the photographer arrange the friends in a row?
To find the number of ways to arrange the friends in a row, we need to use the permutation formula: nPr = n! / (n - r)!. In this case, there are 5 friends, so we need to find the number of permutations of 5 things taken 5 at a time. Therefore, the answer is C. 120.
5. How would you apply the fundamental counting principle to a bike combination with 5 numbers?
The fundamental counting principle states that the total number of outcomes of a sequence of events is equal to the product of the number of ways each event can occur. In this case, we have 5 numbers on a bike combination and each number can be any digit from 0 to 9. Therefore, the number of possible combinations is 10 x 10 x 10 x 10 x 10, or 10^5. Therefore, the answer is C. 10 x 10 x 10 x 10 x 10.
To solve these permutation problems, you can use the formula for permutations:
P(n, r) = n! / (n - r)!
Where n represents the total number of items and r represents the number of items you are selecting or arranging.
Now, let's solve each problem step by step:
1. For the first question, there are 6 books that need to be arranged on a shelf. Since the order matters and no book can repeat, we can apply the permutation formula:
P(6, 6) = 6! / (6 - 6)! = 6! / 0! = 6!
The value of 6! (6 factorial) is equal to 6 × 5 × 4 × 3 × 2 × 1 = 720. Therefore, the answer is d. 720.
2. In the second question, there are 8 girls competing in an ice-skating competition, and we need to determine the number of ways they can finish in first, second, and third place. Again, we can use the permutation formula:
P(8, 3) = 8! / (8 - 3)! = 8! / 5!
Calculating this value:
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
5! = 5 × 4 × 3 × 2 × 1 = 120
Dividing, we get:
P(8, 3) = 40,320 / 120 = 336
Therefore, the answer is a. 336.
3. For the third question, we need to find the number of ways a president and vice president can be selected from a class of 12. Again, we can use the permutation formula:
P(12, 2) = 12! / (12 - 2)! = 12! / 10!
Calculating this value:
12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Dividing, we get:
P(12, 2) = (12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) = 12 × 11 = 132
Therefore, the answer is c. 132.
4. For the fourth question, there are 7 students who need to line up for a class picture. We can use the permutation formula once again:
P(7, 7) = 7! / (7 - 7)! = 7! / 0! = 7!
Calculating this value:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
Therefore, the answer is a. 5,040.
I hope this helps! Let me know if you have any further questions.