Combinations

Combinations are selections made by choosing some or all objects from a set, without considering the order of selection. The number of combinations of n different objects taken r at a time is denoted by \(^nC_r\) and is calculated using the formula:
 

\(^nC_r = \frac{n!}{r!(n-r)!}\), where 0 ≤ r ≤ n
 

This is known as the general combination formula.
 

Let's see an example:
 

A fruit basket contains five different fruits: Apple, Banana, Cherry, Mango, and Orange. In how many ways can you select two fruits from the basket?
 

Solution:
 

\(^5C_2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \, 3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2 \times 1} = 10\)
 

There are 10 different ways to select two fruits from 5.

 

Now that we know what combinations are, let us learn about the formula that we use to easily find the number of possible combinations of objects.

The formula to calculate the number of combinations is:

                C(n,r) = n! / r!(n-r)!

Where:
 

• n is the total number of items
   
• r is the number of items selected (r ≤ n)
   
• !(factorial) is the multiplying of a number by all positive integers below it.

Let us take an example,

If there are 5 different fruits, and you want to pick 2, the number of ways you can select is

C(n,r) = \( \frac{n!}{r!(n - r)!} = C(5,2) = \frac{5!}{2!(5 - 2)!} \)
 

⇒ C(5,2) = \( \frac{5!}{2! \, 3!} = \frac{(5 \times 4 \times 3 \times 2 \times 1)}{(2 \times 1)(3 \times 2 \times 1)} \) = 10

So there are 10 different ways to choose 2 fruits from a total number of 5 fruits.

When learning about combinations, it is important to know whether the order matters or not. To understand this distinction we need to understand the difference between permutations and combinations as it can be quite confusing to know when to use permutations or combinations.
 

Suppose we have a set of 6 letters: {A, B, C, D, E, F}. How many ways can we select a group of 3 letters from this set?

If we first consider arrangements (permutations) of 3 letters from the six letters, the number would be: 

\(^6P_3\)

Now, consider the permutations that contain the letters A, B, and C. There are 3!=6 possible arrangements:

ABC, ACB, BAC, BCA, CAB, CBA

However, for combinations, the order does not matter. All six of these arrangements correspond to a single combination {A, B, C}.

So, to find the total number of combinations of 3 letters from 6, we divide the total permutations by 
3! (the number of ways to arrange three letters):

\(^6C_3 = \frac{^6P_3}{3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\)

This is also read as “6 choose 3.”

Understanding the key points of the combination formula helps explain exceptional cases, such as choosing all, choosing none, and symmetry. These basics make calculations easier and prevent common mistakes.
The combination formula is:

\(^nC_r = \frac{n!}{r!(n-r)!}\) ,where 0 ≤ r ≤ n

The important cases:

1. Where r = n:

\(^nC_n = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \cdot 0!} = 1\)

2. When r = 0:

\(^nC_0 = \frac{n!}{0!(n-0)!} = \frac{n!}{0! \cdot n!} = 1\)

This shows that the combination formula is valid even when r = 0.

3. Symmetry property:

\(^nC_{(n-r)} = \frac{n!}{(n-r)!\,(n-(n-r))!} = \frac{n!}{(n-r)!\,r!} = {}^nC_r\)

This means that selecting r objects from n objects is equivalent to rejecting n−r objects.

Combinations are a difficult topic to understand and solve problems related to it. In this section, we will discuss some tips and tricks to master combinations.

• Understand the concept clearly: Combinations are used when the order of selection does not matter. Focus on “who is chosen,” not “in what order.”
   
• Remember the formula: The basic idea is selecting r items from n items. Knowing the relationship between factorials helps you solve problems faster.
   
• Simplify factorials early: Cancel out common terms in the numerator and denominator to avoid large calculations. This saves time and reduces mistakes.
   
• Practice with small numbers: Start with small, simple examples, like choosing 2 items from 5. It helps you understand how combinations work before tackling bigger problems.
   
• Use the calculator smartly: For larger numbers, use the nCr function on your calculator to check your manual work and save time during exams.
   
• Think in Terms of “Select or Reject”: Encourage children to think in terms of “select or reject” and to break problems into smaller steps rather than memorizing formulas.
   
• Use Real-Life Situations: Use simple, everyday situations like picking team members, choosing fruits, or selecting toys to make combination problems. It would be fun and relatable. Begin with easy tasks and gradually increase the challenge.
   
• Use Games and Activities: Use card games, dice, or board games to show that, in combinations, the order of selection doesn’t matter.
   
• Visualize Small Sets: Give children small sets of objects, 3 to 5 items, and let them list all possible choices. Visual tools like charts, cards, or drawings help them clearly see different combinations.

Combinations are used widely in our daily lives. Here are a few real-world applications of combinations.

Lottery draws: One of the most common uses of combinations, in lotteries a set of numbers is selected. The order in which it is drawn does not matter, this makes it a combination.

Forming sports teams: When selecting players for a team, the order in which they are chosen does not matter, what matters is who is selected.

Card games: When drawing cards for a game, the order of the cards does not matter. What matters is the cards you have. This makes it a combination.

Food and meal choices: Restaurants use combinations to create meal deals or ingredient mixes where order doesn’t matter.

Genetic research: In genetics, combinations are used to study how traits are passed on by analyzing different gene pairings without considering order.