High school math worksheet featuring eight word problems focused on calculating permutations and combinations for various scenarios.

Here are the step-by-step solutions for each problem on the worksheet.

1. 5 members of a school’s 8-person varsity softball team will march in a parade. How many different ways can the players be chosen and lined up?
* Step 1: You need to choose 5 people out of 8, and the order they stand in matters (because they are "lined up"). This is a permutation.
* Step 2: The formula is $P(n, r) = \frac{n!}{(n-r)!}$. Here $n=8$ and $r=5$.
* Step 3: Calculate: $8 \times 7 \times 6 \times 5 \times 4$.
* $8 \times 7 = 56$
* $56 \times 6 = 336$
* $336 \times 5 = 1,680$
* $1,680 \times 4 = 6,720$

2. Mr. White has 5 books, and wants to select 3 to display on his shelf. How many different ways can he arrange the books?
* Step 1: Order matters on a shelf. We are choosing 3 books from 5.
* Step 2: Calculation: $5 \times 4 \times 3$.
* $5 \times 4 = 20$
* $20 \times 3 = 60$

3. 7 members of a school’s 9-person varsity softball team will march in a parade. How many different ways can the players be chosen and lined up?
* Step 1: Choose 7 from 9, order matters.
* Step 2: Calculation: $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3$.
* $9 \times 8 = 72$
* $72 \times 7 = 504$
* $504 \times 6 = 3,024$
* $3,024 \times 5 = 15,120$
* $15,120 \times 4 = 60,480$
* $60,480 \times 3 = 181,440$

4. You have 9 different cats, and want to arrange 3 of them in a single-file line. How many different ways can you arrange the cats?
* Step 1: Choose 3 cats from 9, order matters (single-file line).
* Step 2: Calculation: $9 \times 8 \times 7$.
* $9 \times 8 = 72$
* $72 \times 7 = 504$

5. Mr. White has 8 books, and wants to select 7 to display on his shelf. How many different ways can he arrange the books?
* Step 1: Choose 7 from 8, order matters.
* Step 2: Calculation: $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2$.
* Alternatively, this is just $8!$ (8 factorial) divided by $1!$, which is simply $8!$.
* $8 \times 7 = 56$
* $56 \times 6 = 336$
* $336 \times 5 = 1,680$
* $1,680 \times 4 = 6,720$
* $6,720 \times 3 = 20,160$
* $20,160 \times 2 = 40,320$

6. How many different ways can the letters of the word METRIC be arranged?
* Step 1: Count the letters. M-E-T-R-I-C has 6 distinct letters.
* Step 2: Since all letters are unique, we just calculate 6 factorial ($6!$).
* Step 3: Calculation: $6 \times 5 \times 4 \times 3 \times 2 \times 1$.
* $6 \times 5 = 30$
* $30 \times 4 = 120$
* $120 \times 3 = 360$
* $360 \times 2 = 720$
* $720 \times 1 = 720$

7. Mr. White has 7 books, and wants to select 6 to display on his shelf. How many different ways can he arrange the books?
* Step 1: Choose 6 from 7, order matters.
* Step 2: Calculation: $7 \times 6 \times 5 \times 4 \times 3 \times 2$.
* $7 \times 6 = 42$
* $42 \times 5 = 210$
* $210 \times 4 = 840$
* $840 \times 3 = 2,520$
* $2,520 \times 2 = 5,040$

8. You have 8 different cats, and want to arrange 3 of them in a single-file line. How many different ways can you arrange the cats?
* Step 1: Choose 3 cats from 8, order matters.
* Step 2: Calculation: $8 \times 7 \times 6$.
* $8 \times 7 = 56$
* $56 \times 6 = 336$

Final Answer:
1. 6,720
2. 60
3. 181,440
4. 504
5. 40,320
6. 720
7. 5,040
8. 336