6. A bin of golf tees has 7 different colors. You reach in and grab 100 tees. How many different combinations of tees are possible?
6. A bin of golf tees has 7 different colors. You reach in and grab 100 tees. How many different combinations of tees are possible?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Question 6
**Problem Statement:**
A bin of golf tees has 7 different colors. You reach in and grab 100 tees. How many different combinations of tees are possible?
**Show your work here:**
*Solution:*
To determine how many different combinations of tees are possible, we need to use the concept of combinations with repetition (also known as multisets). This is because we are choosing 100 tees from 7 different colors, and the order does not matter, but repetition of colors is allowed.
The formula for combinations with repetition is given by:
\[
\text{Number of combinations} = \binom{n+r-1}{r}
\]
Where:
- \( n \) is the number of different types (colors) of tees.
- \( r \) is the number of tees to be selected.
In this problem:
- \( n = 7 \) (since there are 7 different colors)
- \( r = 100 \) (since we are selecting 100 tees)
Substitute these values into the formula:
\[
\text{Number of combinations} = \binom{7+100-1}{100} = \binom{106}{100}
\]
Now, calculate \(\binom{106}{100}\):
\[
\binom{106}{100} = \frac{106!}{100!(106-100)!} = \frac{106!}{100! \cdot 6!}
\]
Where \( ! \) denotes factorial. This expression can be computed using either a calculator or computational software to get the exact number. However, factorial calculations can be very large, so often in practice, this would be done using specialized software.
The result is a very large number, indicating the vast number of possible combinations when choosing 100 tees from 7 colors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F826d393f-4b6d-4ba0-9bb7-a16f07f33d0e%2F52c78c5a-e983-4512-a4e6-e3098b7ed556%2F0w4o74_processed.png&w=3840&q=75)
Transcribed Image Text:### Question 6
**Problem Statement:**
A bin of golf tees has 7 different colors. You reach in and grab 100 tees. How many different combinations of tees are possible?
**Show your work here:**
*Solution:*
To determine how many different combinations of tees are possible, we need to use the concept of combinations with repetition (also known as multisets). This is because we are choosing 100 tees from 7 different colors, and the order does not matter, but repetition of colors is allowed.
The formula for combinations with repetition is given by:
\[
\text{Number of combinations} = \binom{n+r-1}{r}
\]
Where:
- \( n \) is the number of different types (colors) of tees.
- \( r \) is the number of tees to be selected.
In this problem:
- \( n = 7 \) (since there are 7 different colors)
- \( r = 100 \) (since we are selecting 100 tees)
Substitute these values into the formula:
\[
\text{Number of combinations} = \binom{7+100-1}{100} = \binom{106}{100}
\]
Now, calculate \(\binom{106}{100}\):
\[
\binom{106}{100} = \frac{106!}{100!(106-100)!} = \frac{106!}{100! \cdot 6!}
\]
Where \( ! \) denotes factorial. This expression can be computed using either a calculator or computational software to get the exact number. However, factorial calculations can be very large, so often in practice, this would be done using specialized software.
The result is a very large number, indicating the vast number of possible combinations when choosing 100 tees from 7 colors.
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