High School

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1. [tex]\frac{41}{4} = 6[/tex]

2. [tex]P(6, 2) \cdot P(5, 3) \div P(3, 2) = 30[/tex]

3. [tex]\frac{10!}{7! \cdot 3!} = 120[/tex]

4. [tex]P(6, r) = 30[/tex] ; [tex]r = 2[/tex]

5. [tex]P(8, 3) = 336[/tex]

6. [tex]P(n, 3) = 60[/tex] ; [tex]n = 5[/tex]

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IV. Read and answer each item carefully. Write the letter that corresponds to your answer on the blank before the number.

11. Annie has 6 different books in her locker. If she takes 4 books from the locker and then arranges them in a row, find the number of permutations for the books.
A. 720
B. 360
C. 24
D. 6

12. How many 3-letter words can be formed from the word ORANGE?
A. 720
B. 120
C. 18
D. 6

13. How many ways can the letters in REARRANGE be arranged?
A. 362,880
B. 15,120
C. 120
D. 9

14. In a fitness center, there are 5 different pieces of equipment. If only 3 pieces of equipment out of 5 are used, how many choices of orders of usage are there?
A. 120
B. 60
C. 20
D. 15

Answer :

Sure! Let's go through each part of the question step-by-step:

1. Expression (8):
Given: [tex]\(\frac{P(6,2) \times P(5,3)}{P(3,2)}\)[/tex]

Permutation formula: [tex]\(P(n, r) = \frac{n!}{(n-r)!}\)[/tex]

- [tex]\(P(6,2) = \frac{6!}{(6-2)!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} = 30\)[/tex]
- [tex]\(P(5,3) = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 60\)[/tex]
- [tex]\(P(3,2) = \frac{3!}{(3-2)!} = \frac{3 \times 2 \times 1}{1} = 6\)[/tex]

Calculation:
[tex]\(\frac{30 \times 60}{6} = \frac{1800}{6} = 300\)[/tex]

2. Expression (4):
Given: [tex]\(\frac{10!}{7! \times 3!}\)[/tex]

Calculation:
[tex]\(\frac{10 \times 9 \times 8 \times 7!}{7! \times 3 \times 2 \times 1} = \frac{720}{6} = 120\)[/tex]

3. Expression (9):
Given: [tex]\(P(6, r) = 30\)[/tex]

Here, we find the value of [tex]\(r\)[/tex]:
- Since [tex]\(P(6, r) = \frac{6!}{(6-r)!} = 30\)[/tex]
- Trying [tex]\(r = 2:\)[/tex]
[tex]\(P(6, 2) = \frac{6!}{4!}\)[/tex]
[tex]\(= \frac{6 \times 5 \times 4!}{4!} = 30\)[/tex]

So, [tex]\(r = 2\)[/tex].

4. Expression (5):
Given: [tex]\(P(8,3) = 336\)[/tex]

- Calculate [tex]\(P(8,3)\)[/tex]:
[tex]\(P(8,3) = \frac{8!}{(8-3)!} = \frac{8 \times 7 \times 6 \times 5!}{5!} = 8 \times 7 \times 6 = 336\)[/tex]

5. Expression (10):
Given: [tex]\(P(n, 3) = 60\)[/tex]

- Find value of [tex]\(n\)[/tex]:
- Trying [tex]\(n = 5\)[/tex]:
[tex]\(P(5,3) = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60\)[/tex]

So, [tex]\(n = 5\)[/tex].

6. Expression (11):
Annie has 6 different books and takes 4 to arrange:

Permutation [tex]\(P(6, 4) = \frac{6!}{(6-4)!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2!} = 6 \times 5 \times 4 \times 3 = 360\)[/tex]

7. Expression (12):
How many 3-letter words from "ORANGE":

- Total letters = 6
- Permutations for 3 letters: [tex]\(P(6,3)\)[/tex]

Calculation:
[tex]\(P(6,3) = \frac{6!}{(6-3)!} = \frac{6 \times 5 \times 4}{1} = 120\)[/tex]

8. Expression (13):
Permutations of "REARRANGE":

REARRANGE has 9 letters, with repetitions: R (3 times), A (2 times).

Calculation:
[tex]\(\frac{9!}{3! \times 2!} = \frac{362,880}{6 \times 2} = \frac{362,880}{12} = 30,240\)[/tex]

9. Expression (14):
Using 3 out of 5 equipment:

- Permutation [tex]\(P(5,3)\)[/tex]

Calculation:
[tex]\(P(5,3) = \frac{5!}{2!} = \frac{5 \times 4 \times 3}{1} = 60\)[/tex]

I hope this helps clarify each calculation step! Let me know if you have any questions.

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