A person has 10 friends in which there are 4 females. If he invites 5 friends from them at random find the probability that (i) 2 females (il) all men, are invited

Let's break down the problem step by step:

### Given:
- Total friends = 10
  - 4 are females
  - \(10 - 4 = 6\) are males.
- He is inviting 5 friends at random.

We need to find the probability for two different cases:
1. **Exactly 2 females are invited.**
2. **All 5 invited are males (i.e., no females are invited).**

### (i) Probability of inviting exactly 2 females:
- To solve this, we need to count how many ways there are to choose exactly 2 females and 3 males (since the total number of invited friends is 5).
- First, calculate how many ways to select 2 females from 4:
  \[
  \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
  \]
- Next, calculate how many ways to select 3 males from 6:
  \[
  \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
  \]
- Therefore, the total number of favorable outcomes is:
  \[
  \binom{4}{2} \times \binom{6}{3} = 6 \times 20 = 120
  \]

- Now, calculate the total number of possible ways to select 5 friends from 10:
  \[
  \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
  \]

- Thus, the probability of inviting exactly 2 females is:
  \[
  P(\text{2 females}) = \frac{120}{252} = \frac{40}{84} = \frac{20}{42} = \frac{10}{21}
  \]

### (ii) Probability of inviting all men (no females):
- Here, we need to select all 5 friends from the 6 males, so the number of favorable outcomes is:
  \[
  \binom{6}{5} = \frac{6}{1} = 6
  \]
- The total number of possible selections remains \( \binom{10}{5} = 252 \).

- Therefore, the probability of inviting all males is:
  \[
  P(\text{all men}) = \frac{6}{252} = \frac{1}{42}
  \]

### Final Answer:
- (i) Probability of inviting exactly 2 females: \( \frac{10}{21} \).
- (ii) Probability of inviting all men (no females): \( \frac{1}{42} \).

Would you like more details or have any questions? 

Here are some related questions for practice:
1. What is the probability of inviting at least 1 female?
2. What is the probability of inviting exactly 3 females?
3. How would the answer change if the total number of friends was 12, with 5 females?
4. What is the probability of inviting more females than males?
5. If 4 friends are invited, what is the probability that exactly 1 female is included?

**Tip:** Always verify your combination formulas to ensure you are choosing the correct number of elements.

A person has 10 friends in which there are 4 females. If he invites 5 friends from them at random find the probability that (i) 2 females (ii) all men, are invited.

Explore how to calculate the probability of inviting exactly 2 females or all males when randomly selecting 5 friends from a group of 10, including 4 females and 6 males. This problem involves combinatorics and probability theory, suitable for students in Grades 10-12.

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