A delegation of ve students is to be
selected for a Model United Nations
conference. There are 10 boys and 13 girls
to choose from.
a In how many different ways can a
delegation be chosen if there are no
restrictions?
b If the team is to include at least one girl
and one boy, in how many ways can the
delegation be selected?

### Part (a)
In this part, you are asked to determine how many different ways a delegation of 5 students can be selected from a total of 23 students (10 boys and 13 girls) without any restrictions. 

The formula for selecting a group of \( r \) objects from \( n \) total objects (without regard to the order) is given by the combination formula:

\[
\binom{n}{r} = \frac{n!}{r!(n - r)!}
\]

In this case, \( n = 23 \) (total boys and girls) and \( r = 5 \) (delegation size). Thus, the number of ways to select a delegation of 5 students is:

\[
\binom{23}{5} = \frac{23!}{5!(23 - 5)!} = \frac{23!}{5!18!}
\]

Now calculating the value of this expression:

\[
\binom{23}{5} = \frac{23 \times 22 \times 21 \times 20 \times 19}{5 \times 4 \times 3 \times 2 \times 1} = 33649
\]

Thus, there are **33,649 ways** to select the delegation without any restrictions.

---

### Part (b)
Now, if the team must include at least **one boy** and **one girl**, we need to exclude the cases where the team is made up of only boys or only girls.

1. **Number of ways to select a team with only boys**:
   Here, we choose all 5 students from the 10 boys. The number of ways to do this is:

   \[
   \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
   \]

2. **Number of ways to select a team with only girls**:
   Similarly, we choose all 5 students from the 13 girls. The number of ways to do this is:

   \[
   \binom{13}{5} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287
   \]

3. **Total teams with no restrictions**:
   This was calculated in part (a) as 33,649.

4. **Teams with only boys or only girls**:
   The total number of such teams is \( 252 + 1287 = 1539 \).

5. **Teams with at least one boy and one girl**:
   Subtract the number of teams with only boys or only girls from the total number of teams:

   \[
   33649 - 1539 = 32110
   \]

Thus, there are **32,110 ways** to select the delegation if it must include at least one boy and one girl.

---

Do you want more details or have any questions?

### Related Questions:
1. What if the delegation size was 6 students instead of 5?
2. How would the calculation change if at least 2 boys and 2 girls were required?
3. What if there was an upper limit of 3 girls in the delegation?
4. How many ways can you select a delegation of all boys?
5. How many ways can you select a delegation with exactly 3 boys and 2 girls?

### Tip:
Always double-check the conditions of the problem. Ensure that any special restrictions (such as requiring both boys and girls) are accounted for before finalizing the calculation.

A delegation of five students is to be selected for a Model United Nations conference. There are 10 boys and 13 girls to choose from. a) In how many different ways can a delegation be chosen if there are no restrictions? b) If the team is to include at least one girl and one boy, in how many ways can the delegation be selected?

Learn how to calculate the number of ways to select a delegation of five students from 10 boys and 13 girls, with or without restrictions. This combinatorics problem covers key concepts such as combinations and the counting principle, suitable for high school students in Grades 9-12.

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