True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
6. 7C5=7C2

Verified step by step guidance

1. Step 1: Recall the formula for combinations, which is used to calculate the number of ways to choose r items from a set of n items. The formula is: $C(n,r)=\frac{n!}{r!(n-r)!}$.
2. Step 2: Apply the formula to calculate $C_{7}5$. Substitute n = 7 and r = 5 into the formula: $\frac{7!}{5!(7-5)!}$.
3. Step 3: Simplify the denominator of the formula for $C_{7}5$. The denominator becomes $5!(2!)$, since $7-5=2$.
4. Step 4: Similarly, calculate $C_{7}2$ using the same formula. Substitute n = 7 and r = 2 into the formula: $\frac{7!}{2!(7-2)!}$.
5. Step 5: Observe that the calculations for $C_{7}5$ and $C_{7}2$ are equivalent because of the symmetry property of combinations: $C_{n}r=C_n(n-r)$. Conclude that the statement is true.

Key Concepts

Combinatorial Notation

Combinatorial notation, often represented as nCr or C(n, r), denotes the number of ways to choose r elements from a set of n elements without regard to the order of selection. It is calculated using the formula nCr = n! / (r!(n-r)!), where '!' denotes factorial, the product of all positive integers up to that number.

Factorial

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating combinations and permutations, as they help determine the total arrangements or selections possible.

Properties of Combinations

One important property of combinations is that C(n, r) = C(n, n-r). This means that choosing r elements from n is equivalent to leaving out n-r elements. This property is crucial for simplifying combinatorial expressions and understanding relationships between different combinations.