Question

Sunil has 6 friends. In how many ways can he invite two or more of his friends for dinner? (1) 58 (2) 57 (3) 63 (4) 49

Short answer

Answer: 57 ways

Step-by-step solution

Using the Combination Formula

The combination formula is used to find the number of ways to choose 'r' elements from a set of 'n' elements, where the order of the elements does not matter. The formula is as follows: C(n,r) = n! / (r!(n-r)!) Where C(n,r) denotes the number of combinations, n! is the factorial of n, and r! is the factorial of r.

Calculate the combinations for 2, 3, 4, 5 and 6 friends

Using the combination formula, we will calculate the number of ways to invite different sets of friends: C(6,2) = 6! / (2!(6-2)!) = 15 C(6,3) = 6! / (3!(6-3)!) = 20 C(6,4) = 6! / (4!(6-4)!) = 15 C(6,5) = 6! / (5!(6-5)!) = 6 C(6,6) = 6! / (6!(6-6)!) = 1

Find the total number of ways

To find the total number of ways Sunil can invite two or more friends, we will add the combinations found in step 2: Total number of ways = C(6,2) + C(6,3) + C(6,4) + C(6,5) + C(6,6) = 15 + 20 + 15 + 6 + 1 = 57 So, Sunil can invite two or more of his friends for dinner in 57 ways. The correct answer is (2) 57.

Key concepts

Combination Formula

When dealing with problems that involve selecting groups of items, you often use the combination formula. This formula helps determine how many ways you can choose a subset of items from a larger set. Here, the order doesn't matter. The mathematical formula for combinations is:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]where:

• \(n\) is the total number of items.
• \(r\) is the number of items you want to choose.
• \(!\) represents factorial, which is a product of all positive integers up to that number.

In our exercise, Sunil has 6 friends and wants to find out the different ways to invite at least 2 of them for dinner. By using the combination formula, he calculates options for inviting groups of 2, 3, 4, 5, and even all 6 friends. This method covers all possible combinations where the group count is 2 or more.

Factorials

Factorials are a crucial part of the combination formula. The factorial of a number \(n\), represented as \(n!\), is the product of all positive integers from 1 up to \(n\). For example:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

It's important to note that \(0!\) is defined as 1. Factorials grow very quickly with larger numbers, which influences the denominator and numerator in the combination formula. In Sunil's exercise, we rely on these calculations to properly execute the combination formula when determining the number of invitations.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arranging, and analyzing finite discrete structures. It's essential in problems involving combinations and permutations. While permutations consider the order of selection, combinations don't.
The problem Sunil faces is a classical example of combinatorics, where he needs to find out in how many different ways he can create groups of friends to invite for dinner. By using the concept of combinations, he can solve this without having to list or calculate every possible group manually.
Combinatorics is widely used in many fields, such as computer science for data analysis, statistics, and even gaming for calculating probabilities.

Binomial Coefficients

Binomial coefficients are numerical factors in the expansion of binomials raised to a power, often seen in the binomial theorem. It is expressed as \(C(n, r)\) or often written as \(\binom{n}{r}\).
These coefficients can be used to determine the number of ways to pick \(r\) items from \(n\) items, similar to the combination formula. In the case of Sunil's problem, each combination such as \(\binom{6}{2}\) or \(\binom{6}{3}\) represents a binomial coefficient.
These coefficients are symmetrical, meaning \(\binom{n}{r} = \binom{n}{n-r}\), which helps simplify calculations, particularly in larger data sets. Hence, understanding binomial coefficients enhances our ability to quickly and efficiently solve selection and distribution problems.