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 :

Options

A. a. `8/39`
B. b. `15/39`
C. c. `12/13`
D. d. none of these

Answer

To solve the problem of finding the probability that at least one couple is selected when 4 persons are picked from 8 couples (16 individuals), we can follow these steps:

### Step 1: Understand the total number of people and couples
We have 8 couples, which means there are a total of 16 individuals (8 husbands and 8 wives).

### Step 2: Calculate the total ways to select 4 people from 16
The total number of ways to select 4 people from 16 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{16}{4} \]
Calculating this:
\[ \binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820 \]

### Step 3: Calculate the ways to select 4 people without selecting any couple
To ensure that no couple is selected, we can only select from the 8 husbands or the 8 wives. Thus, we can select 4 individuals from 8 (either all husbands or all wives):
\[ \text{Ways to select 4 people from 8} = \binom{8}{4} \]
Calculating this:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 4: Calculate the probability of selecting no couples
The probability of selecting 4 people without selecting any couple is given by the ratio of the number of ways to select 4 people without couples to the total number of ways to select 4 people:
\[ P(\text{no couples}) = \frac{\text{Ways to select 4 without couples}}{\text{Total ways to select 4}} = \frac{70}{1820} \]
Simplifying this:
\[ P(\text{no couples}) = \frac{1}{26} \]

### Step 5: Calculate the probability of selecting at least one couple
The probability of selecting at least one couple is the complement of the probability of selecting no couples:
\[ P(\text{at least one couple}) = 1 - P(\text{no couples}) = 1 - \frac{1}{26} = \frac{25}{26} \]

### Final Answer
Thus, the probability that at least one couple will be selected is:
\[ \frac{25}{26} \]

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