There are 4 girls and 6 boys in a class.
(a)
In how many ways can a student be selected?
(b)
In how many ways a team of 1 girl and 1 boy can be selected?
(c) In how many ways a project group of 4 students consisting of at least 1 girl can be selected?
(1 mark)
(2 marks)
(5 marks)
(d) Now the class is going to take a photo,
(i) in how many ways can they be seated in a row?
(ii) in how many ways can they be seated in a row that all the girls are always
together?

Let’s solve each part step-by-step.

### Given:
- 4 girls and 6 boys in a class.

---

### 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:
\[
10 \text{ ways}
\]

---

#### (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:
\[
4 \times 6 = 24 \text{ ways}
\]

---

#### (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.

---

#### (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:
   \[
   7! \times 4! = 5,040 \times 24 = 120,960 \text{ ways}
   \]

---

### 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?

---

### 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.

There are 4 girls and 6 boys in a class.
(a) In how many ways can a student be selected?
(b) In how many ways a team of 1 girl and 1 boy can be selected?
(c) In how many ways a project group of 4 students consisting of at least 1 girl can be selected?
(1 mark)
(2 marks)
(5 marks)
(d) Now the class is going to take a photo,
(i) in how many ways can they be seated in a row?
(ii) in how many ways can they be seated in a row that all the girls are always together?

Learn how to solve a combinatorial problem involving selection and arrangement of students in class. Includes solutions for selecting teams with specific constraints and arranging students with conditions. This exercise covers combinations, permutations, and the counting principle, suitable for Grades 10-12.

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