# What is the Minimum Number of Friends Needed for Unique Dinner Invitations?

• MHB
• Lancelot1
In summary, the conversation is about solving a problem involving inviting friends to dinner where John has n friends and wants to invite three each day for 365 days without repeating the same group. The solution involves using the inequality $\binom{n}{3}\ge365$ and finding the lower bound of n to be the smallest natural number equal to or larger than the root of the equation, which is 14.
Lancelot1
Hello all,

I am trying to solve this one:

John has n friends . He wants to invite in each evening (365 days a year) three of his friends for dinner. What should be the size of n, such that it will be possible not to invite the same triplet twice ?

What I did was:

$\binom{n}{3}\leq 365$

which turns into:

$\frac{(n-1)(n-2)n}{6}\leq 365$

I have tried to solve it, manually and with a mathematical software, in both ways n was not a natural number...where is my mistake ?

Lancelot said:
I have tried to solve it, manually and with a mathematical software, in both ways n was not a natural number
First, the inequality should be $\binom{n}{3}\ge365$. Second, the answer to this problem is an inequality $n\ge\ldots$, not a specific value of $n$. For the lower bound of $n$ take the smallest natural number that is equal to or larger than the root of that equation. Note that $\binom{14}{3}=364$.

## 1. What is a binomial coefficient?

A binomial coefficient is a mathematical expression that represents the number of ways to choose a subset of objects from a larger set. It is often denoted by the symbol "n choose k" and is calculated using the formula n! / (k! * (n-k)!), where n is the total number of objects and k is the number of objects in the subset.

## 2. How is a binomial coefficient used in probability?

In probability, a binomial coefficient is used to calculate the probability of a specific number of successes in a series of independent trials. It is often used in binomial experiments, where there are only two possible outcomes (success or failure) and the probability of success remains constant for each trial.

## 3. What is the connection between binomial coefficients and Pascal's Triangle?

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The coefficients in the expansion of (a + b)^n form the nth row of Pascal's Triangle, with the first number being the binomial coefficient for n choose 0 and the last number being the binomial coefficient for n choose n.

## 4. How are binomial coefficients used in combinatorics?

In combinatorics, binomial coefficients are used to count the number of ways to arrange or choose objects in a specific order. They are also used to calculate the number of permutations and combinations of a given set of objects.

## 5. Can binomial coefficients be negative?

No, binomial coefficients cannot be negative. They represent the number of ways to choose a subset of objects, which is always a positive value. If the result of the calculation is negative, it means that the objects cannot be chosen in that specific way.

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