Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Verified step by step guidance

1. Identify the problem as a combination problem because the order of selection does not matter. We need to find the number of ways to choose 6 numbers from 53 without regard to order.
2. Recall the formula for combinations, which is given by: $\mathrm{nCr}=\frac{n!}{r!\; (n-r)!}$, where $n$ is the total number of items, and $r$ is the number of items to choose.
3. Substitute the given values into the formula: $n=\; 53$ and $r=\; 6$. So, the expression becomes $\backslash binom\{53\}\{6\}\; =\; \backslash frac\{53!\}\{6!\; (53-6)!\}$.
4. Simplify the factorial expression by expanding only the necessary terms to avoid calculating large factorials completely. For example, write $53!\; =\; 53\; \backslash times\; 52\; \backslash times\; 51\; \backslash times\; 50\; \backslash times\; 49\; \backslash times\; 48\; \backslash times\; 47!$ and cancel $47!$ in numerator and denominator.
5. Calculate the numerator product and divide by $6!$ to find the total number of different selections possible.

Key Concepts

Combination Formula (nCr)

The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for problems where order does not matter, such as selecting lottery numbers.

Factorials

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in the combination formula to count permutations and combinations by accounting for the number of ways to arrange or select items.

Order Irrelevance in Combinations

In combinations, the order of selection does not matter, meaning that selecting numbers {1, 2, 3} is the same as {3, 2, 1}. This contrasts with permutations, where order matters. Understanding this distinction helps determine when to use combinations versus permutations in counting problems.