Problem 37 - Combinatorics Through Guided Discovery

Question

In how many ways can we pass out \(k\) (identical) ping-pong balls to \(n\) children if each child may get at most one? (h)

Step-by-Step Solution

Answer

\(\binom{n}{k}\)

Step 1: Understand the problem

The problem requires distributing a maximum of 1 ping-pong ball to each child such that each child gets at most one ball and there are a total of k ping-pong balls and n children.

Step 2: Consider the conditions

Each child can either receive 1 ping-pong ball or none. The total number of ways to distribute these balls has to satisfy the condition that no child gets more than one ball.

Step 3: Determine the number of children and balls

Identify the number of children, n, and the number of ping-pong balls, k.

Step 4: Apply combinations formula

Since each child can get a maximum of one ball, the problem boils down to selecting k children out of n to give the balls. The number of ways to choose k children out of n is given by the binomial coefficient: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Key Concepts

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics. It represents the number of ways to choose a subset of items from a larger set, without considering the order of the items. Mathematically, it is denoted as \(\binom{n}{k}\), and it reads as 'n choose k'. This is computed using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(!\) denotes the factorial operator. The binomial coefficient tells you how many different ways you can select \(k\) balls from \(n\) children, assuming each child can receive at most one ball. It simplifies the distribution problem and makes it manageable using straightforward arithmetic calculations.

Combinatorial Distribution

Combinatorial distribution deals with the strategies to distribute items among groups following specific rules. In our problem, we are distributing \(k\) identical ping-pong balls to \(n\) children, each child getting at most one ball. The key here is that once the positions for the balls are chosen (which children will receive the balls), the actual distribution is straightforward because each child can get only one ball or none. This means we need to count the combinations rather than permutations. The binomial coefficient \(\binom{n}{k}\) gives us the exact number of ways to accomplish this distribution.

Factorials

Factorials play a significant role in combinatorics and are crucial to understanding how to calculate the binomial coefficient. The factorial of a positive integer \(n\), denoted as \(n! \), is the product of all positive integers less than or equal to \(n\). For example, \(!5 = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow very quickly with increasing \(n\), so while the calculations can become large, they simplify expressions in combinatorial formulas. When you compute \(\binom{n}{k}\), factorials help manage the different ways of choosing subsets, ensuring both the numerator and denominator account for all the possible selections and arrangements.