Question

There are 9 subjects in eighth standard but there are only 6 periods in a day. In how many ways can the time table be formed? (a) 60480 (b) 23460 (c) 56780 (d) 61280

Short answer

a) 60480 b) 109440 c) 45360 d) 75600 Answer: a) 60480 Explanation: Using the permutation formula with n = 9 subjects and r = 6 periods, we calculate P(9,6) = 9! / (9-6)! = 362880 / 6 = 60480.

Step-by-step solution

Identify the values of n and r

In this problem, there are 9 subjects (n) that can be placed into 6 periods (r).

Plug the values into the permutation formula

The permutation formula is P(n,r) = n! / (n-r)!. We plug in n = 9 and r = 6 to get P(9,6) = 9! / (9-6)!.

Calculate the factorials

Now, we need to calculate the factorials: 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362880 (9-6)! = 3! = 3 × 2 × 1 = 6

Calculate the number of ways

We'll now divide 9! by 3! to obtain the number of ways the time table can be formed: P(9,6) = 362880 / 6 = 60480

Choose the correct option

The number of ways to form the time table is 60480, so the correct answer is (a) 60480.

Key concepts

Factorials

Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. The factorial of a number, denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). This concept is not only useful in permutations but also appears in many areas of science and engineering.
For example:

• \(0! = 1\) (by definition)
• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• and so on...

Factorials grow very quickly, which is why they are particularly suitable for solving problems involving arrangements or sequences, such as the timetable creation problem. By calculating \(9!\) and dividing by \((9-6)!\), we determined the number of permutations for arranging 9 subjects during 6 periods.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of sets of elements. It's instrumental for problems that involve the arrangement of groups, such as creating a schedule from a set of subjects in this exercise.
The permutation formula, \(P(n, r) = \frac{n!}{(n-r)!}\), is used when we need to find how many different ways we can arrange \(r\) elements out of a total \(n\) without repetition. This particular formula helps when the order in which we arrange the subjects matters, as with creating a timetable.
In this exercise, we calculated permutations by using the formula with \(n = 9\) and \(r = 6\), showcasing the application of combinatorics in finding efficient solutions to arrangement problems. It's a crucial concept for tasks like data arrangement, optimizing schedules, and many more real-world applications.

Mathematics Education

Mathematics education focuses on teaching concepts such as permutations and factorials in a way that students can easily grasp and apply them. Real-world examples, like scheduling subjects into periods, help illustrate abstract mathematical ideas.
Effective mathematics teaching involves breaking down complex formulas into simple steps, such as learning how to calculate factorials or understanding combinatorics. Contextual problems, similar to the timetable creation problem, enable students to see the application of math in everyday life.
The step-by-step approach in the exercise illustrates how tackling a complex problem can be managed by first identifying data, applying formulas, calculating necessary operations, and interpreting results. This approach not only aids in solving mathematical problems but also develops critical-thinking skills in students.