Cavities in Maths: Class 10 Mathematics: CBSE Sample Question Paper (2020-21) (Basic) - 4

Introduction:
In mathematics, "cavities" generally refers to the concept of finding the number of empty spaces or gaps within a given pattern or arrangement. It is a term commonly used in combinatorics and counting principles to determine the number of possibilities or arrangements by identifying the vacant positions or gaps.

Understanding Cavities:
Cavities can be visualized as the empty spaces or slots in a given pattern or arrangement. These empty spaces play a crucial role in determining the number of possible combinations or arrangements.

Example:
Let's consider a simple example to understand the concept of cavities. Suppose we have a sequence of numbers: 1, 2, 3, 4, 5. We want to find the number of possible three-digit numbers that can be formed using these digits without repetition.

Process:
1. We start by identifying the vacant positions or cavities in a three-digit number. In this case, we have three cavities or empty spaces.
2. For the first cavity, we have five options to choose from (1, 2, 3, 4, or 5).
3. After selecting a digit for the first cavity, we are left with four remaining digits.
4. For the second cavity, we have four options to choose from (the remaining digits after selecting one for the first cavity).
5. After selecting a digit for the second cavity, we are left with three remaining digits.
6. For the third and final cavity, we have three options to choose from (the remaining digits after selecting one for the first and second cavities).
7. Finally, we multiply the number of options for each cavity: 5 * 4 * 3 = 60.

Conclusion:
In conclusion, cavities in mathematics refer to the vacant positions or empty spaces within a given pattern or arrangement. By identifying these cavities, we can determine the number of possible combinations or arrangements. It is an important concept in combinatorics and counting principles, helping us solve problems involving permutations and combinations.