Question

Winning a state lottery is based on trying to guess which six randomly picked numbers in the set \(\\{1,2, \ldots, 30\\}\) will be chosen. No repeats are allowed. Winning a second state lottery is based on trying to guess which six randomly picked numbers from the set \(\\{1,2, \ldots, 38\\}\) will be chosen. Winning a third state lottery is based on trying to guess 7 of 11 randomly picked numbers from the set of \(\\{1,2, \ldots, 80\\}\). How many possible winning combinations are there for each of these lotteries?

Short answer

593775; 2760681; 23602537520

Step-by-step solution

Determine combinations for the first lottery

For the first lottery, calculate how many combinations of 6 numbers can be picked from a set of 30 numbers. The formula for combinations is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 30 \) and \( r = 6 \). Calculate \( \binom{30}{6} = \frac{30!}{6!(30-6)!} \). Start by simplifying the factorials:\[ \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]This yields 593775 combinations.

Determine combinations for the second lottery

For the second lottery, calculate how many combinations of 6 numbers can be picked from a set of 38 numbers. Use the same formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 38 \) and \( r = 6 \).Calculate \( \binom{38}{6} = \frac{38!}{6!(38-6)!} \).Simplify the factorials:\[ \frac{38 \times 37 \times 36 \times 35 \times 34 \times 33}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]This yields 2760681 combinations.

Determine combinations for the third lottery

For the third lottery, calculate how many ways to choose 7 numbers out of the 11 selected from a set of 80. First find combinations of selecting 11 numbers from 80, then choose 7 from these 11: 1. Calculate the combinations of selecting 11 numbers from 80: \( \binom{80}{11} = \frac{80!}{11!(80-11)!} \).Simplify the computation by the formula:\[ \frac{80 \times 79 \times \cdots \times 70}{11!} \].This results in 71523144 combinations to choose 11 numbers.2. For choosing 7 of these 11 numbers:\( \binom{11}{7} = \frac{11!}{7!(11-7)!} \).This equals \( \binom{11}{4} \), which is 330 combinations.Multiply these two results:\( 71523144 \times 330 = 23602537520 \) combinations.

Key concepts

Lottery Combinations

When trying to understand the concept of lottery combinations, it's essential to know that the lottery is a classic example of combinatorics. In a lottery, you're asked to pick a certain number of numbers from a larger set, and you win by guessing the correct combination. Let's break this down a bit: - **Combinations**: Unlike permutations, combinations focus on selecting items where the order doesn't matter. For instance, choosing numbers 1, 2, 3 is the same as choosing 3, 2, 1 in a combination. - **No Repetition Allowed**: In lotteries, each number can only be picked once, adding an extra challenge. For example: - **First Lottery**: Selecting 6 numbers from 30. - **Second Lottery**: Selecting 6 numbers from 38. - **Third Lottery**: Selecting 7 numbers out of a previously chosen 11 from 80. The challenge in these problems is calculating the total number of possible combinations.

Factorials

Factorials are at the heart of calculating combinations in lotteries. A factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). For example, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). Here's how they work in lottery combinations:

• Factorials simplify the calculation by reducing large computations to more manageable forms.
• The factorial for any number grows very quickly and can become astronomically large, thus it's often simplified in a fraction format.

For instance, in the first step to calculate how many combinations exist for a lottery involving 30 numbers, we use:The combination formula, which includes factorials:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] Using factorials like \( 30! \) helps break the problem into simpler multiplication tasks. This factorial representation is crucial in understanding the complex breakdown of the numbers involved.

Binomial Coefficient

The binomial coefficient, often noted as \( \binom{n}{r} \), is a powerful tool used to calculate combinations. It tells us how many ways we can choose \( r \) items from a total of \( n \) items, without considering the order of those items. Here's why it's important:- **Formula**: \( \binom{n}{r} = \frac{n!}{r! \cdot (n-r)!} \)- **Purpose**: It provides a systematic way to determine how many possible combinations can be made from a selection pool.In the context of lotteries:- In the first and second lotteries, the binomial coefficient helps to determine all possible sets of 6 numbers from 30 and 38 respectively.- In the third lottery, it’s used twice: first to find how many ways you can choose 11 numbers from 80, and then to pick 7 numbers from those 11.The binomial coefficient simplifies these calculations, allowing for a logical approach to unraveling seemingly complex number problems. It's a shorthand that captures the essence of the problem without wading through every possible permutation.