Question 7

You need to find how many ways at least two defective batteries can be included in the purchase of 7 batteries from a total of 40, where 3 are defective.

This is a problem of combinations, with a focus on "at least two defective" batteries.

1. Total number of ways to choose 7 batteries (without any restriction): $\text{Total combinations} = \binom{40}{7}$

2. Number of ways with no defective batteries (choosing all 7 from the 37 non-defective ones): $\text{No defective} = \binom{37}{7}$

3. Number of ways with exactly 1 defective battery (choose 1 defective from 3, and 6 non-defective from 37): $\text{1 defective} = \binom{3}{1} \times \binom{37}{6}$

4. The number of ways with at least 2 defective batteries will be: $\text{At least 2 defective} = \binom{40}{7} - \left( \binom{37}{7} + \binom{3}{1} \times \binom{37}{6} \right)$

Question 8

You need to select 5 students out of 18 boys and 20 girls, where more boys than girls are selected. The possible selections are:

• 3 boys, 2 girls
• 4 boys, 1 girl
• 5 boys, 0 girls

For each case, calculate the combinations:

1. 3 boys, 2 girls: $\binom{18}{3} \times \binom{20}{2}$

2. 4 boys, 1 girl: $\binom{18}{4} \times \binom{20}{1}$

3. 5 boys, 0 girls: $\binom{18}{5} \times \binom{20}{0}$

Add all of these up to get the total number of ways to select more boys than girls.

Question 9

The problem asks for the number of ways to select at least one junior partner in a committee of 3 from a group of 25 senior and 26 junior partners.

1. Total number of ways to form the committee without restrictions: $\text{Total combinations} = \binom{51}{3}$

2. Number of ways to select only senior partners (choosing all 3 from 25): $\text{Only senior} = \binom{25}{3}$

3. Number of ways to have at least one junior partner: $\text{At least 1 junior} = \binom{51}{3} - \binom{25}{3}$

Would you like detailed steps for any of the questions or have any other queries?

Here are 5 relative questions to practice:

1. How do you compute combinations involving larger groups?
2. What is the general formula for solving "at least" type problems?
3. How does inclusion-exclusion principle help in counting?
4. How would you approach probability with similar counting principles?
5. How do factorials relate to permutations and combinations?

Tip: For large combination calculations, simplify factorials by canceling out terms to reduce computational complexity.