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.