Question

Choose the correct option from the given options below by solving the following question:
In a group of $8$ boys, two boys are brothers. The number of ways in which they can sit in a row if the brothers are not to sit together is:
A. $30240$
B. $1410$
C. $2830$
D. $8420$

Answer 1

Find the total number of combinations, then the combinations in which the brothers can sit together. Subtract it from the total number of combinations to get the number of combinations in which the brothers will not sit together.

Complete step-by-step solution:
Total number of boys in a group $= 8$
Given condition,
The brothers will not sit in a row.
We have to find the total number of ways in which the boys can sit in a row with the brothers not sitting together.
So, we can write the condition like,
The number of ways in which the brothers will not sit together $=$ the total number of ways $-$ the number of ways in which the brothers will sit together.
Now,
We shall write each term stated above mathematically using combinations.
The total number of ways $= 8!$
Now, expanding the factorial, we get;
$8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Multiplying the terms, we get;
Total number of ways $= 4032$
The number of ways in which the two brothers will sit together $= 7!$
Here, we consider the two brothers as one element since they sit together, but they can change their place along the seven seats.
So, we have, $7! \times 2$
Expanding the factorial, we get;
$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 2$
The number of ways in which the two brothers will sit together$= 10080$
Now,
We apply these substituted values in the equation we got at the first, i.e.,
The number of ways in which the brothers will not sit together $=$ the total number of ways $-$ the number of ways in which the brothers will sit together, we get;
The number of ways in which the brothers will not sit together $=$ $40320 - 10080$
$\Rightarrow$ The number of ways in which the brothers will not sit together $= 30240$

The correct option is A.

Note: The factorial function in mathematics is a positive integer followed by an exclamation, i.e., it is represented in the form of $n!$. This function represents the product of all the positive integers less than $n$. The value of $0!$ being an exception because the value of $0!$ is unitary, i.e., one. This factorial function is mainly used in permutations and combinations.