Evaluate the given expression and express the result using the usual format for writing numbers (instead of scientific notation). 30 C3 30 C3 =
Evaluate the given expression and express the result using the usual format for writing numbers (instead of scientific notation). 30 C3 30 C3 =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Combinatorics Evaluation
**Problem Statement:**
Evaluate the given expression and express the result using the usual format for writing numbers (instead of scientific notation).
**Given Expression:**
\[ ^{30}C_{3} \]
**Question:**
Calculate \(^{30}C_{3}\)
**Answer:**
\[^{30}C_{3} = \boxed{} \]
### Explanation:
The given problem is a combinatorics problem where we need to calculate the number of ways to choose 3 items from a set of 30 items.
\[ ^{30}C_{3} \] represents a combination, which can be calculated using the formula:
\[ ^{n}C_{r} = \frac{n!}{r!(n-r)!} \]
For this particular problem:
\[ n = 30 \]
\[ r = 3 \]
Therefore:
\[ ^{30}C_{3} = \frac{30!}{3!(30-3)!} = \frac{30!}{3! \cdot 27!} \]
Next, simplify the factorials:
\[ 30! = 30 \times 29 \times 28 \times 27! \]
So:
\[ \frac{30!}{3! \cdot 27!} = \frac{30 \times 29 \times 28 \times 27!}{3! \times 27!} = \frac{30 \times 29 \times 28}{3!} \]
Since \( 3! = 6 \):
\[ ^{30}C_{3} = \frac{30 \times 29 \times 28}{6} \]
Calculating the values:
\[ 30 \times 29 = 870 \]
\[ 870 \times 28 = 24360 \]
Finally, divide by 6:
\[ \frac{24360}{6} = 4060 \]
So, the answer is:
\[ ^{30}C_{3} = \boxed{4060} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb955d04e-ca8a-4cb8-88af-517f600cb5dc%2F0491bf6b-4d84-47a0-b911-ec459a741bea%2Fxs1cmn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Combinatorics Evaluation
**Problem Statement:**
Evaluate the given expression and express the result using the usual format for writing numbers (instead of scientific notation).
**Given Expression:**
\[ ^{30}C_{3} \]
**Question:**
Calculate \(^{30}C_{3}\)
**Answer:**
\[^{30}C_{3} = \boxed{} \]
### Explanation:
The given problem is a combinatorics problem where we need to calculate the number of ways to choose 3 items from a set of 30 items.
\[ ^{30}C_{3} \] represents a combination, which can be calculated using the formula:
\[ ^{n}C_{r} = \frac{n!}{r!(n-r)!} \]
For this particular problem:
\[ n = 30 \]
\[ r = 3 \]
Therefore:
\[ ^{30}C_{3} = \frac{30!}{3!(30-3)!} = \frac{30!}{3! \cdot 27!} \]
Next, simplify the factorials:
\[ 30! = 30 \times 29 \times 28 \times 27! \]
So:
\[ \frac{30!}{3! \cdot 27!} = \frac{30 \times 29 \times 28 \times 27!}{3! \times 27!} = \frac{30 \times 29 \times 28}{3!} \]
Since \( 3! = 6 \):
\[ ^{30}C_{3} = \frac{30 \times 29 \times 28}{6} \]
Calculating the values:
\[ 30 \times 29 = 870 \]
\[ 870 \times 28 = 24360 \]
Finally, divide by 6:
\[ \frac{24360}{6} = 4060 \]
So, the answer is:
\[ ^{30}C_{3} = \boxed{4060} \]
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