Question

Six married couples are sitting in a room. Number of ways in which 4 people can be selected so that there is exactly one married couple among the four is N,N is a three digit number of the form abc then $(a + b - c)$ is equal to

Answer 1

According to the question we have to find $(a + b - c)$ when six married couples are sitting in a room. Number of ways in which 4 people can be selected so that there is exactly one married couple among the four is N,N is a three digit number of the form abc.
So, first of all as given in the question there are 6 couples means the total number of members are 12 but according to the questions 4 people are selected so we have to choose the 2 couples or 4 people from the total number of couples or people.

Formula used: $n_r^c = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}.................(1)$
After finding the pair of couples now we have to add both of the cases when only one couple is selected and when two couples are selected. Now after comparing the total number of ways with the given expression $(a + b - c)$ we can find the value of the given expression.

Complete step-by-step answer:
Step 1: The number of couples given in the question is 6, so the number of total people is 12.
Step 2: Now, 4 people are selected so that means 2 couples are selected so, first of all we have to
find the total number of ways when there is only one couple selected with the help of the formula
(1) as mentioned in the solution hint.
$= c_1^6 \times c_2^5 \times 2 \times 2$
On solving the obtained expression with the help of the formula (1),
$= \left( {\dfrac{{6!}}{{1!\left( {6 - 1} \right)!}}} \right) \times \left( {\dfrac{{5!}}{{2!\left( {5 - 2} \right)!}}} \right) \times 2 \times 2 \\\ = \left( {\dfrac{{6!}}{{5!}}} \right) \times \left( {\dfrac{{5!}}{{2!3!}}} \right) \times 4 \\\ = \dfrac{{6 \times 5!}}{{5!}} \times \dfrac{{5 \times 4 \times 3!}}{{2!3!}} \times 4 \\\ = 6 \times 40 \\\ = 240 \\\$
Step 3: Now, we have to choose the 2 couples from the given 6 couples with the help of the formula (1) as mentioned in the solution hint.
$= c_2^6 \\\ = \dfrac{{6!}}{{2!\left( {6 - 2} \right)!}} \\\ = \dfrac{{6!}}{{2!4!}} \\\ = \dfrac{{6 \times 5 \times 4!}}{{2!4!}} \\\ = 15 \\\$
Step 4: Now, we have to add the both of the cases when only one and two couples are selected in
the step 2 and step 3.
$= 240 + 15 \\\ = 255 \\\$
Now, on comparing a = 2 , b = 5, and c = 5
Step 5: Now, on substituting the value of a, b, and c we can obtain the value of the given expression
$(a + b - c)$
$= 2 + 5 - 5 \\\ = 2 \\\$

Hence, with the help of the formula (1) we have obtained the value of the given expression $(a + b - c)$= 2

Note: To find the value of 6! or 5! We have to use the method to find the value of the factorial of a given number n, hence n! =n(n-1)(n-2)(n-3)………………2$\times$1
The factorial of a number is the product of all the integers from 1 to that given number.