# In how many ways can a teacher arrange 10 students in the front row if there are 25 total students?

In mathematics, permutation refers to the act of organizing a set in which all the members of a set are arranged into some sequence or order. In other words, if the set is already arranged, then the rearranging of its component is called the process of permuting. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

**Permutation Formula**

A permutation is the choice of r things from a set of n things without replacement and where the sequence matters.

^{n}P_{r} = (n!)/(n-r)!

**What is a Combination?**

A combination is an act of choosing items from a group, such that (unlike permutations) the order of choice does not matter. In smaller cases, it is possible to count the number of amalgamations. Combination refers to the amalgamation of n things taken k at a time without repetition. To refer to combinations in which reoccurrence is permitted, the terms k-selection or k-combination with repetition are frequently used.

**Combination Formula**

A combination is the choice of r things from a set of n things without replacement and where order does not matter.

**In how many ways can a teacher arrange 10 students in the front row if there are 25 total students?**

**Solution:-**

Here in this problem, we have to determine the total number of ways in which we can

arrange some number of students out of certain students in the front row of the

classroom. First of all, we will determine the total number of ways in which we can

select 10 students out of 25 number of students. Then we will permute or arrange them.

No. of ways of selecting r things out of n is = (n¦r)

The number of ways in which we can arrange r number of different things = r!

Answer and Explanation:

Total number of students = 25

Number of places in the front row = 10

Number of ways in which 10 students can be arranged in a row is 10!. This is because

each student can be seated in any of the 10 seats. There could be 3628800 possible

arrangements.

Number of ways selecting 10 students out of 25 = (25¦10)

**Similar Questions**

**Question 1: There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. ****In how many ways can the committee be formed?**

**Answer:-**

We need to select 5 men from 8 men and 6 women from 10 women.

Number of ways to do this

=

^{8}C_{5}×^{10}C_{6}=

^{8}C_{3}×^{10}C_{4}[∵^{n}C_{r}=^{n}C_{(n-r)}]= [(8 × 7 × 6)/(3 × 2 × 1)] × [(10 × 9 × 8 × 7)/(4 × 3 × 2 × 1)]

= 56 × 210

= 11760

**Question 2: In how many ways can a team of 5 persons be formed out of a total of 10 persons such that two particular persons should be included in each team?**

**Answer:**

Two particular persons should be included in each team. Therefore we have to select the

remaining (5 – 2) = 3 persons from (10 – 2) = 8 persons.

Hence, the required number of ways

=

^{8}C_{3}= (8×7×6)/(3×2×1)

= 8 × 7

= 56