Solved Problems

Problem 1: You are at an ice cream parlor, and they offer 10 different flavors of ice cream. You can choose 3 scoops of ice cream. How many different combinations of ice cream can you order?

Solution:

This is a combination problem. You want to find 10 choose 3 (¹⁰C₃).

¹⁰C₃ = 10! / (3!(10-3)!) = 120 different combinations.

So, there are 120 different ways to order 3 scoops of ice cream from 10 flavors.

Problem 2: In a school with 30 students, there are 5 positions available on the student council: president, vice-president, secretary, treasurer, and historian. How many different ways can the positions be filled?

Solution:

This is a permutation problem because the order of election to the positions matters.

For the president, there are 30 choices.

For the vice-president, there are 29 choices (since one person is already president).

For the secretary, there are 28 choices (after president and vice-president are selected).

For the treasurer, there are 27 choices.

For the historian, there are 26 choices.

Now, multiply these choices together to get the total number of ways to fill the positions:

30 × 29 × 28 × 27 × 26 = 17,956,800 different ways to fill the positions.

So, there are 17,956,800 different ways to elect the student council.

nCr Formula is one of the countless formulas in the world of mathematics, which plays a pivotal role in solving problems and gaining a deeper understanding of the subject. nCr formula deals with combinations and as we know, Combinations are an integral part of combinatorics, the branch of mathematics that focuses on counting and arranging objects.

They are widely used in probability and statistics to calculate the possible outcomes of events. They also have many applications in real-life situations such as forming teams, choosing passwords, arranging books, etc. In this article, we will explore the nCr formula in detail, discussing its importance, and applications and providing clarity through solved problems.