Problem Analysis and Solution
The image provides a worksheet on permutations, where the order of selection matters. The two main formulas for permutations are:
1.
Repetition Allowed: \( nPr = n^r \)
2.
Repetition Not Allowed: \( nPr = \frac{n!}{(n-r)!} \)
We will solve each of the four examples using these formulas.
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Example 1: How many ways can the letters in the word "MAINE" be arranged if you use all 5 letters (and repetition is allowed)?
####
Solution:
- The word "MAINE" has 5 distinct letters.
- Since repetition is allowed, we use the formula \( nPr = n^r \).
- Here, \( n = 5 \) (the number of letters to choose from) and \( r = 5 \) (we are arranging all 5 letters).
\[
nPr = 5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125
\]
####
Answer:
\[
\boxed{3125}
\]
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Example 2: How many different 3-digit codes can you make with the single digit whole numbers (without repeating)?
####
Solution:
- The single digit whole numbers are \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \), so \( n = 10 \).
- We need to form a 3-digit code without repetition, so \( r = 3 \).
- Since repetition is not allowed, we use the formula \( nPr = \frac{n!}{(n-r)!} \).
\[
nPr = \frac{10!}{(10-3)!} = \frac{10!}{7!} = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 = 720
\]
####
Answer:
\[
\boxed{720}
\]
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Example 3: How many ways can 3 people win 1st, 2nd, and 3rd place?
####
Solution:
- There are 3 positions to fill: 1st, 2nd, and 3rd place.
- Since the order matters and repetition is not allowed, we use the formula \( nPr = \frac{n!}{(n-r)!} \).
- Here, \( n = 3 \) (the number of people) and \( r = 3 \) (we are selecting all 3 people).
\[
nPr = \frac{3!}{(3-3)!} = \frac{3!}{0!} = \frac{3 \times 2 \times 1}{1} = 6
\]
####
Answer:
\[
\boxed{6}
\]
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Example 4: How many ways can 8 students sit in 4 seats?
####
Solution:
- We have 8 students and 4 seats, so we need to arrange 4 students out of 8.
- Since the order matters and repetition is not allowed, we use the formula \( nPr = \frac{n!}{(n-r)!} \).
- Here, \( n = 8 \) (the number of students) and \( r = 4 \) (the number of seats).
\[
nPr = \frac{8!}{(8-4)!} = \frac{8!}{4!} = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4!} = 8 \times 7 \times 6 \times 5 = 1680
\]
####
Answer:
\[
\boxed{1680}
\]
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Final Answers:
1. \(\boxed{3125}\)
2. \(\boxed{720}\)
3. \(\boxed{6}\)
4. \(\boxed{1680}\)
Parent Tip: Review the logic above to help your child master the concept of permutation and combination worksheet.
