Question

8 couples (husband and wife) attend a dance show 'Nach Baliye' in a popular TV channel ; A lucky draw in which 4 persons picked up for a prize is held, then the probability that there is atleast one couple will be selected is : (a) \(\frac{8}{39}\) (b) \(\frac{15}{39}\) (c) \(\frac{12}{13}\) (d) none of these

Short answer

a) $\frac{6}{35}$ b) $\frac{27}{91}$ c) $\frac{12}{15}$ d) None of these Answer: d) None of these (The probability is $\frac{35}{91}$)

Step-by-step solution

Calculate the total ways to select 4 persons from the 16 attendees

We have a total of 16 individuals (8 couples) to choose from, and we must choose 4 people to receive the prize. This can be represented by the binomial coefficient \(C(16,4)\) which can be calculated as: $$ C(16,4) = \frac{16!}{(16-4)! \times 4!} = \frac{16!}{12! \times 4!} = 1820 $$ This means there are 1820 different ways to select 4 individuals from the 16 attendees.

Calculate the ways to select 4 persons without selecting a couple

To calculate the number of ways to select 4 persons without selecting a couple, we need to pick one person from each of four different couples. We have 8 couples to choose from, so there are \(C(8,4)\) ways to select four different couples. Then, from each selected couple, we can choose one person. This gives us a total of \(C(8,4) \times 2^4\) ways to select 4 individuals without selecting a couple: $$ C(8,4) \times 2^4 = 70 \times 16 = 1120 $$

Calculate the probability that no couple is selected

To find the probability that no couple is selected, we divide the number of ways to select 4 persons without selecting a couple (1120) by the total number of ways to select 4 persons from the 16 attendees (1820): $$ P(\text{no couple}) = \frac{1120}{1820} = \frac{56}{91} $$

Find the complementary probability

Since the problem asks for the probability of at least one couple being selected, we need to find the complementary probability, which is the probability that at least one couple is selected: $$ P(\text{at least one couple}) = 1 - P(\text{no couple}) = 1 - \frac{56}{91} = \frac{35}{91} $$ The probability of at least one couple being selected is \(\frac{35}{91}\). The given options do not match this probability, so the correct answer is (d) none of these.

Key concepts

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and finding patterns within certain structures or sets. It allows us to solve problems related to the arrangement of objects without having to list out all possibilities, which would be impractical or impossible for large sets. Within combinatorics, we utilize various methods and principles, like the 'Rule of Sum' and the 'Rule of Product', to find the number of possible combinations and permutations.

When solving probability problems involving combinatorics, it's essential to identify the type of scenario we're dealing with - whether it's a matter of combination, where the order of selection doesn't matter, or permutation, where the order is significant. In the exercise provided, we use combinations because the order in which couples are selected for the prize doesn't affect the outcome.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics used to count the number of ways in which a set of objects can be arranged. Permutations focus on arrangements where the order matters, such as lining up people in a queue, whereas combinations deal with groupings where the order doesn't matter, like selecting committee members from a larger group.

To distinguish between the two, remember that combinations, denoted by the symbol 'C', are used when the order of selection is irrelevant. Permutations, represented by 'P', are chosen when the order is important. The textbook exercise we are discussing requires the use of combinations since the selection order of the dance show attendees for the lucky draw prize is not significant. By understanding this distinction, students can correctly approach a wide range of problems and calculate the number of possible outcomes.

Binomial Coefficient

The binomial coefficient, commonly expressed in the form of 'C(n, k)' or often read as 'n choose k', represents the number of combinations of 'n' items taken 'k' at a time. This concept is pivotal in calculating probabilities and combinatorial figures where grouping without regard to order is the focus. The formula for the binomial coefficient is:\[\begin{equation}C(n, k) = \frac{n!}{k!(n-k)!}\end{equation}\]
Here, 'n!' denotes the factorial of 'n', which is the product of all positive integers up to 'n'. The binomial coefficient is crucial in diverse areas ranging from algebra to probability theory.

In the context of the exercise, the binomial coefficient helped us determine the total number of ways to select 4 people from 16 attendees. Correct application of these coefficients enables understanding of how many configurations can be made, a fundamental step in solving probability-based combinatorial problems.