Evaluate each expression.
$\frac{{}_7{C}_3}{{}_5{C}_4}-\frac{98!}{96!}$

Verified step by step guidance

1. Identify the combinations in the expression: \$7C3\( and \)5C4$. Recall that the combination formula is given by \(nCr = \frac{n!}{r!(n-r)!}\).
2. Write out each combination using the formula: \(7C3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\) and \(5C4 = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!}\).
3. Calculate each combination separately by simplifying the factorial expressions. For example, simplify \$7!\(, \)3!\(, and \)4!\( as needed to find \)7C3\(, and similarly for \)5C4$.
4. Evaluate the factorial fraction \(\frac{98!}{96!}\) by simplifying it. Remember that \(\frac{98!}{96!} = 98 \times 97\) because the factorial terms cancel out except for the last two factors.
5. Substitute the values of \$7C3\(, \)5C4$, and \(\frac{98!}{96!}\) back into the original expression and perform the subtraction to complete the evaluation.

Key Concepts

Combination Formula

A combination represents the number of ways to choose a subset of items from a larger set without regard to order. It is calculated using the formula nCr = n! / [r!(n-r)!], where n is the total number of items and r is the number chosen.

Factorials

A factorial, denoted by n!, is the product of all positive integers from 1 up to n. Factorials are fundamental in permutations and combinations, simplifying expressions involving counting and arrangements.

Simplifying Factorial Expressions

When evaluating expressions involving factorials, it is often helpful to cancel common terms in the numerator and denominator. This simplification reduces computational complexity and helps in accurately calculating combinations or other factorial-based expressions.