Let’s solve each part step-by-step.

Given:

• 4 girls and 6 boys in a class.

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Solution:

(a) In how many ways can a student be selected?

There are $4$ girls and $6$ boys, so a total of: $4 + 6 = 10 \text{ students}$ Since we need to select only one student, the answer is: $10 \text{ ways}$

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(b) In how many ways can a team of 1 girl and 1 boy be selected?

• We have 4 choices for selecting 1 girl (since there are 4 girls).
• We have 6 choices for selecting 1 boy (since there are 6 boys).

Using the counting principle: $4 \times 6 = 24 \text{ ways}$

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(c) In how many ways can a project group of 4 students consisting of at least 1 girl be selected?

To form a group of 4 students with at least 1 girl, we can proceed as follows:

1. Calculate the total number of ways to select any 4 students from 10 students: $\binom{10}{4} = 210$

2. Calculate the number of ways to select 4 students with no girls (only boys):

   • There are only 6 boys, so the number of ways to select 4 boys from 6 is: $\binom{6}{4} = 15$

3. Subtract the all-boys selection from the total selection: $210 - 15 = 195$

Thus, there are $195$ ways to form a group of 4 students with at least 1 girl.

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(d) Now the class is going to take a photo:

(i) In how many ways can they be seated in a row?

There are 10 students in total, so they can be seated in: $10! = 3,628,800 \text{ ways}$

(ii) In how many ways can they be seated in a row so that all the girls are always together?

If all 4 girls sit together as a single “block,” then:

1. We can think of the 4 girls as one unit. This gives us 7 units to arrange in a row (the "block" of girls + 6 boys).

   The 7 units can be arranged in: $7! = 5,040 \text{ ways}$

2. Within the "block" of 4 girls, they can be arranged among themselves in: $4! = 24 \text{ ways}$

3. Total arrangements: $7! \times 4! = 5,040 \times 24 = 120,960 \text{ ways}$

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Summary of Answers:

1. (a) 10 ways
2. (b) 24 ways
3. (c) 195 ways
4. (d)(i) 3,628,800 ways
5. (d)(ii) 120,960 ways

Would you like further explanation on any part?

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Related Questions:

1. How many ways can a team of 2 girls and 2 boys be selected?
2. In how many ways can a project group of 5 students with at least 2 girls be chosen?
3. What if two students must always sit together in the photo? How many arrangements are possible?
4. How would the solution change if we needed a group of exactly 3 girls?
5. In how many ways can two boys be selected from the group?

Tip:

When calculating ways with conditions (like "at least one girl"), consider using complements for efficiency.