Textbook Question
Use the formula for nCr to evaluate each expression. 9C5
564
views
1
rank
10. Combinatorics & Probability
Combinatorics
Problem 10
Textbook Question
Use the formula for nCr to evaluate each expression. 10C6
Verified step by step guidance1
Recall the formula for combinations, which is used to find the number of ways to choose \(r\) objects from \(n\) objects without regard to order:
\[ \text{nCr} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Identify the values of \(n\) and \(r\) from the problem. Here, \(n = 10\) and \(r = 6\).
Substitute these values into the formula:
\[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} \]
Simplify the factorial expressions by expanding only as much as needed to cancel terms. For example, write \$10!\( as \(10 \times 9 \times 8 \times 7 \times 6!\) so that \)6!$ cancels out in numerator and denominator.
After canceling, you will have a fraction with multiplication in numerator and denominator. Multiply the numbers in numerator and denominator separately, then divide to find the value of \(\binom{10}{6}\).
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combination Formula (nCr)
The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial.
Recommended video:
5:22
Combinations
Factorials
A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in computing combinations and permutations.
Recommended video:
5:22Factorials
Simplifying Factorial Expressions
When evaluating combinations, simplifying factorial expressions by canceling common terms in numerator and denominator helps reduce calculation complexity. For example, 10! / 6! can be simplified by expanding only necessary terms.
Recommended video:
5:22Factorials
Related Videos
Related Practice