Combination

COMBINATION

Chapter 9 of the sources, titled Combination, focuses on the mathematical methods for selecting objects when the order of selection does not matter . This is the primary distinction from permutations, which are used when order is significant.

1. Definition and Fundamental Formula

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Combination Formula ( $nCr$ )

A combination is a selection of $r$ objects from a collection of $n$ distinct objects. The notation used is $nCr$ or $\binom{n}{r}$ , known as the binomial coefficient.

$nCr = \frac{n!}{r!(n-r)!}$

Relationship to Permutation: Each combination of $r$ objects can be arranged in $r!$ ways; therefore, $nCr = \frac{nPr}{r!}$ .

2. Key Mathematical Results

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Symmetry Property

$nCr = nC(n-r)$ Selecting $r$ objects to keep is the same as rejecting $n-r$ objects.

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Boundary Values

For any value of $n$ :

$nCn = 1$
$nC0 = 1$

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Pascal's Rule

$nCr = {}^{n-1}C_{r-1} + {}^{n-1}C_r$ (Provided that $1 \leq r \leq n$ )

3. Permutation vs. Combination

Understanding when to use which method is a core topic:

Permutation

Use when order matters .

Example: Awarding Gold, Silver, and Bronze medals.

Combination

Use when order does not matter .

Example: Choosing the top three athletes to proceed to a next round.

4. Drawing Lines and Shapes

Combinations are used to solve geometric counting problems:

Lines in a Circle: Given $n$ points on a circle, the number of unique line segments (without direction) is $nC2$ .
Triangles: To determine the number of triangles that can be formed from $n$ points where no three are collinear, you use $nC3$ .

Practice Session

Example 1: Selection Logic

How many ways can we select two students from a group of three ( $A, B, C$ )?

$3C2 = \frac{3!}{2!(3-2)!} = \mathbf{3}$ ways ( $AB, AC, BC$ )

Example 2: Grouping with Constraints

A team of 11 from 17 players. If 5 are bowlers and the team must include exactly 4 bowlers?

Select 4 bowlers from 5: $5C4 = 5$
Select 7 others from 12: $12C7 = 792$
Total: $5 \times 792 = \mathbf{3960}$ ways

Solving for $n$

If $nC2 = nC3$ , find the value of $n$ .

View Detailed Solution ▼

Using the property $nCx = nCy \implies x + y = n$ , we have: $2 + 3 = n \implies \mathbf{n = 5}$

Committee Formation

Out of a group of six men and eight women, you need to form a committee of three men and five women. In how many ways can this be done?

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Select 3 men from 6: $6C3 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$ .
Select 5 women from 8: $8C5 = 8C3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$ .
Total: $20 \times 56 = \mathbf{1120}$ ways.

Playing Cards

In how many ways can you choose 4 cards from a 52-card deck if all four must be of the same suit?

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Choose 1 suit out of 4: $4C1 = 4$ ways.
Choose 4 cards from the 13 in that suit: $13C4 = 715$ ways.
Total: $4 \times 715 = \mathbf{2860}$ ways.

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The Grocery Bag Analogy

Think of a Combination as a grocery bag.

When you pick three items—an apple, a banana, and a pear—and put them in your bag, it doesn’t matter which one you grabbed first; you still have the same bag of fruit.

A Permutation, however, is like a priority list for eating them. If your list says “Apple first, then Banana, then Pear,” that is a completely different experience from “Pear first, then Apple, then Banana.”

In combinations, we only care about what is in the bag.