Problem 94 - Precalculus with Limits

Question

In Exercises 91 - 94, prove the identity. \( _nC_r = \dfrac{_nP_r}{r!} \)

Step-by-Step Solution

Answer

The identity \(_nC_r = \frac{_nP_r}{r!}\) is correct as per the formulas used for combinations and permutations.

Step 1 - Understand the formula for combinations

The number of combinations of n items taken r at a time, denoted \(_nC_r\), is given by the formula: \(_nC_r = \frac{n!}{r!(n-r)!}\), where \(n!\) is the factorial of n, which represents the product of all positive integers from 1 to n, \(r!\) is the factorial of r, and \((n-r)!\) is the factorial of (n-r).

Step 2 - Understand the formula for permutations

The number of permutations of n items taken r at a time, denoted \(_nP_r\), is given by the formula: \(_nP_r = \frac{n!}{(n-r)!}\). Permutations differ from combinations in that order matters. For example, AB is considered a different permutation than BA, but the same combination.

Step 3 - Substitute permutation formula into the identity

Now, we will substitute the formula for permutation \(_nP_r\) in the original identity: \[ _nC_r = \frac{_nP_r}{r!} \] which becomes \[ _nC_r = \frac{ \frac{n!}{(n-r)!} }{r!} \] Simplifying this gives: \[ _nC_r = n!/(r!(n-r)! \]

Step 4 - Compare both formulas

It is evident now that the formulas for the combination and the right-hand side of the identity are the same, so the identity \(_nC_r = \frac{_nP_r}{r!}\) is indeed correct.

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics. They represent the product of all positive integers from 1 up to a particular number, say n, and it is denoted by \( n! \). For instance, the factorial of 5, written as \( 5! \), is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It’s a simple yet powerful tool for counting various arrangements and combinations.

Factorials are helpful in various mathematical calculations because they can succinctly represent the number of ways to arrange n items. As such, they are also used in defining formulas for both permutations and combinations. Knowing how to calculate a factorial is key to understanding more complex combinatorial concepts.

Combinations

Combinations are used when you want to find out how many ways you can choose a subset of items from a larger set, where the order does not matter. The formula for combinations of n items taken r at a time is given by \( _nC_r = \frac{n!}{r!(n-r)!} \).

This means you use the factorial of the total number of items \( n! \), divided by the factorial of the number of items you're choosing \( r! \), and the factorial of the remaining items \((n-r)!\). Remember, the essential aspect of combinations is that the sequence in which you select the items does not matter. For example, selecting {A, B} is considered the same as selecting {B, A}.

Understanding combinations is useful for solving problems involving selections, like picking teams from a group or choosing toppings on a pizza.

Permutations

Permutations are all about arranging items where the order does matter. The formula for the number of permutations of n items taken r at a time is \( _nP_r = \frac{n!}{(n-r)!} \).

Unlike combinations, permutations are concerned with the sequence of the selection. For instance, with the items A and B, permutations would identify AB and BA as distinct sequences. This allows permutations to be used in scenarios where the arrangement or order is crucial, such as in seating arrangements, queue formations, or even password settings.

By using permutations, you can calculate how many ways you can order the items, which broadens the scope of how you analyze and resolve problems. This deeper understanding bridges the formula link between combinations and permutations, evident in the identity \( _nC_r = \frac{_nP_r}{r!} \). By rearranging the permutation formula, it's easy to see how it aligns seamlessly with the combination formula.