What are Combinations?

S7-SA3-0383

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Combinations are ways of selecting items from a larger group where the order of selection does not matter. Think of it as forming a team or choosing ingredients for a dish; the sequence in which you pick them doesn't change the final set.

Simple Example

Quick Example

Imagine you have 3 different fruits: Apple, Banana, and Chikoo. You want to pick 2 fruits to make a fruit salad. If you pick Apple then Banana, it's the same salad as picking Banana then Apple. The possible combinations are (Apple, Banana), (Apple, Chikoo), and (Banana, Chikoo). There are 3 combinations.

Worked Example

Step-by-Step

PROBLEM: A school cricket coach needs to select 3 players for a special batting drill from a group of 5 talented players (let's call them P1, P2, P3, P4, P5). How many different groups of 3 players can he select?

STEP 1: Identify the total number of items (n) and the number of items to choose (r).
Here, n = 5 (total players) and r = 3 (players to select).

---STEP 2: Recall the formula for combinations: C(n, r) = n! / (r! * (n-r)!)

---STEP 3: Substitute the values into the formula.
C(5, 3) = 5! / (3! * (5-3)!)

---STEP 4: Simplify the expression.
C(5, 3) = 5! / (3! * 2!)

---STEP 5: Expand the factorials.
5! = 5 * 4 * 3 * 2 * 1 = 120
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2

---STEP 6: Substitute the expanded factorials back into the formula.
C(5, 3) = 120 / (6 * 2)
C(5, 3) = 120 / 12

---STEP 7: Calculate the final answer.
C(5, 3) = 10

ANSWER: The coach can select 10 different groups of 3 players.

Why It Matters

Combinations help us count possibilities in many real-world situations, from designing experiments in Biotechnology to understanding probabilities in FinTech. Knowing combinations can help you design efficient algorithms in AI/ML or even plan routes for delivery services.

Common Mistakes

MISTAKE: Confusing combinations with permutations, thinking order matters. | CORRECTION: Remember, for combinations, the order of selection does NOT change the group. If you pick A then B, it's the same as B then A.

MISTAKE: Incorrectly calculating factorials, especially for larger numbers. | CORRECTION: Always expand factorials carefully (e.g., 4! = 4 x 3 x 2 x 1). Remember that 0! = 1.

MISTAKE: Forgetting to divide by r! in the combination formula, essentially using the permutation formula instead. | CORRECTION: The division by r! in C(n,r) = n! / (r! * (n-r)!) is crucial to remove the duplicates caused by different orderings.

Practice Questions

Try It Yourself

QUESTION: From a group of 7 friends, how many ways can you choose 2 friends to go for a movie? | ANSWER: C(7, 2) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21 ways.

QUESTION: A class has 10 students. The teacher wants to form a committee of 3 students. How many different committees can be formed? | ANSWER: C(10, 3) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120 committees.

QUESTION: You have 8 different colours of rangoli powder. How many ways can you choose 4 colours to make a design? If you must include your favourite colour (let's say red), how many ways can you choose the 4 colours now? | ANSWER: Total ways to choose 4 from 8 = C(8, 4) = 8! / (4! * 4!) = (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1) = 70 ways. If red is a must, you need to choose 3 more colours from the remaining 7 colours. So, C(7, 3) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35 ways.

MCQ

Quick Quiz

What is the value of C(6, 2)?

The Correct Answer Is:

C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 30 / 2 = 15. The other options are incorrect calculations.

Real World Connection

In the Real World

In cricket, when the selection committee picks 11 players for the Indian team from a pool of 20, they are using combinations. The order they announce the names doesn't change the final team. Similarly, when you choose toppings for your pizza or ingredients for a chai blend, you're looking at combinations.

Key Vocabulary

Key Terms

Factorial: The product of an integer and all the integers below it (e.g., 5! = 5x4x3x2x1) | Selection: The act of choosing items from a group | Order: The sequence in which items are arranged or chosen | Set: A collection of distinct objects | Group: A collection of items or individuals

What's Next

What to Learn Next

Next, you should explore 'Permutations'. Understanding permutations will help you see the key difference between situations where order matters and where it doesn't, building a stronger foundation for probability.