Evaluate each expression.
$\frac{{}_7{P}_3}{3!}{-}_7{C}_3$

Verified step by step guidance

1. Identify the components of the expression: \(\frac{^7P_3}{3!} - 7C_3\). Here, \(^7P_3\) represents the number of permutations of 7 items taken 3 at a time, \$3!\( is the factorial of 3, and \)7C_3$ is the number of combinations of 7 items taken 3 at a time.
2. Recall the formulas for permutations and combinations: - Permutations: \(^nP_r = \frac{n!}{(n-r)!}\) - Combinations: \(^nC_r = \frac{n!}{r!(n-r)!}\)
3. Calculate \(^7P_3\) using the permutation formula: \(^7P_3 = \frac{7!}{(7-3)!} = \frac{7!}{4!}\).
4. Calculate \$3!$ which is the factorial of 3: \(3! = 3 \times 2 \times 1\).
5. Calculate \$7C_3$ using the combination formula: \(7C_3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\).

Key Concepts

Permutation

A permutation refers to the arrangement of objects in a specific order. The notation nPr represents the number of ways to arrange r objects from a set of n distinct objects, calculated as n! / (n-r)!. Understanding permutations is essential when order matters in selection.

Combination

A combination is a selection of objects where order does not matter. The notation nCr represents the number of ways to choose r objects from n distinct objects, calculated as n! / [r!(n-r)!]. Combinations are used when the arrangement order is irrelevant.

Factorial and Simplification of Expressions

Factorials (n!) represent the product of all positive integers up to n and are fundamental in calculating permutations and combinations. Simplifying expressions involving factorials, permutations, and combinations requires careful manipulation to reduce complex terms and evaluate the expression correctly.