What is nCr?

S3-SA1-0176

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

nCr is a way to find out how many different groups you can make from a larger set of items, when the order of the items in the group doesn't matter. It helps us count combinations, like choosing a team from players without caring who was picked first.

Simple Example

Quick Example

Imagine you have 5 different flavours of ice cream: Vanilla, Chocolate, Strawberry, Mango, and Pista. You want to choose 2 scoops for your kulfi. How many different pairs of flavours can you pick? nCr helps you find this number.

Worked Example

Step-by-Step

Let's say you have 4 friends (A, B, C, D) and you want to choose 2 of them to come to your birthday party. How many different pairs of friends can you invite?

Step 1: Identify 'n' and 'r'. 'n' is the total number of items (4 friends). 'r' is the number of items you want to choose (2 friends).
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Step 2: Recall the formula for nCr: nCr = n! / (r! * (n-r)!)
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Step 3: Calculate n! (n factorial). 4! = 4 * 3 * 2 * 1 = 24.
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Step 4: Calculate r! (r factorial). 2! = 2 * 1 = 2.
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Step 5: Calculate (n-r)! (n minus r factorial). (4-2)! = 2! = 2 * 1 = 2.
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Step 6: Plug these values into the formula: nCr = 24 / (2 * 2).
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Step 7: Simplify the calculation: nCr = 24 / 4 = 6.
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Answer: You can invite your friends in 6 different pairs.

Why It Matters

nCr is super important in many fields! Data scientists use it to select samples for analysis, and engineers use it in designing systems. It's also key in understanding probability, which is used in everything from weather prediction to game development.

Common Mistakes

MISTAKE: Confusing nCr with nPr (Permutations), where order matters. | CORRECTION: Remember, in nCr (combinations), the order of selection does NOT matter. Choosing 'mango then pista' is the same as 'pista then mango'.

MISTAKE: Incorrectly calculating factorials, especially for (n-r)!. | CORRECTION: Always calculate n!, r!, and (n-r)! separately and carefully before plugging them into the formula.

MISTAKE: Forgetting to divide by r! in the nCr formula. | CORRECTION: The nCr formula has r! in the denominator to remove duplicate orderings, unlike nPr. Always include it!

Practice Questions

Try It Yourself

QUESTION: You have 6 different colours of pens. How many ways can you choose 3 pens to put in your pencil box? | ANSWER: 20 ways

QUESTION: A cricket team has 15 players. How many different groups of 11 players can be chosen for a match? (Hint: 15! is a very big number, so simplify fractions where possible) | ANSWER: 1365 groups

QUESTION: A pizza shop offers 8 different toppings. If you want to choose 4 toppings for your pizza, and one of them MUST be paneer, how many different combinations of toppings can you make? | ANSWER: 35 combinations

MCQ

Quick Quiz

What does 'r' represent in the nCr formula?

The total number of items available

The number of items to be chosen

The result of the calculation

The order of items

The Correct Answer Is:

In nCr, 'n' is the total number of items, and 'r' is the number of items you are selecting from that total. 'C' stands for combinations.

Real World Connection

In the Real World

When you pick your favourite songs to create a playlist on Spotify or JioSaavn, and the order doesn't matter, you're essentially making a combination! Similarly, when a chef selects a few ingredients from a list to create a new dish, nCr helps count the possibilities.

Key Vocabulary

Key Terms

COMBINATION: A selection of items where the order does not matter | FACTORIAL: The product of an integer and all the integers below it (e.g., 5! = 5x4x3x2x1) | n: Total number of items available | r: Number of items to be chosen

What's Next

What to Learn Next

Great job understanding nCr! Next, you should explore 'What is nPr?' (Permutations). This will help you understand situations where the order of selection DOES matter, building on what you've learned about combinations.