Textbook Question
Use the formula for nPr to evaluate each expression. 10P4
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10. Combinatorics & Probability
Combinatorics
Problem 14
Textbook Question
Use the formula for nCr to evaluate each expression. 4C4
Verified step by step guidance1
Recall the formula for combinations, which is used to find the number of ways to choose r objects from a set of n objects without regard to order: \[ \displaystyle {n \choose r} = \frac{n!}{r! (n-r)!} \]
Identify the values of n and r from the problem: here, \[ n = 4 \] and \[ r = 4 \].
Substitute the values of n and r into the combination formula: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! (4-4)!} \].
Simplify the factorial expressions in the denominator: \[ (4-4)! = 0! \], and recall that by definition, \[ 0! = 1 \].
Write the simplified expression ready for evaluation: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! \times 1} \].
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combination Formula (nCr)
The combination formula 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. This formula is fundamental for solving problems involving selections or groups.
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Factorial Function
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in permutations and combinations to count arrangements and selections systematically.
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Evaluating Special Cases in Combinations
When r equals n in nCr, the combination equals 1 because there is exactly one way to choose all items from the set. Understanding these special cases helps simplify calculations and avoid unnecessary computation.
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