34. Lottery Number Selection A lottery has 52 numbers. In how many different ways can six of the numbers be selected? (Assume the order of selection is not important.)

Verified step by step guidance

1. Step 1: Recognize that this is a combination problem because the order of selection does not matter. The formula for combinations is given by: $C=\frac{n}{!}$, where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
2. Step 2: Identify the values of n and r from the problem. Here, n = 52 (total numbers in the lottery) and r = 6 (numbers to be selected).
3. Step 3: Substitute the values of n and r into the combination formula: $C=\frac{\mathrm{52}}{!}$.
4. Step 4: Simplify the factorials in the denominator. Compute (52 - 6) = 46, so the formula becomes: $C=\frac{\mathrm{52}}{!}$.
5. Step 5: Cancel out the common terms in the numerator and denominator. This simplifies to: $C=\frac{\mathrm{52}\times \mathrm{51}\times \mathrm{50}\times \mathrm{49}\times \mathrm{48}\times \mathrm{47}}{6}$. Compute this to find the total number of ways.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this context, we are interested in choosing 6 numbers from a total of 52, which is a classic example of a combination problem. The formula for combinations is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in calculating combinations and permutations, as they help determine the total arrangements of a set of items. In the lottery question, factorials will be used to compute the number of ways to choose 6 numbers from 52.

Binomial Coefficient

The binomial coefficient, often represented as C(n, k) or 'n choose k', quantifies the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, k) = n! / (k!(n-k)!). This concept is crucial for solving the lottery number selection problem, as it directly provides the number of combinations of 6 numbers from 52.