Certainly! Let's break down the problem step-by-step to find the number of possible combinations:
### (a) Selecting 2 boys and 1 girl
1. Number of boys and girls:
- We have 8 boys and 4 girls.
2. Number of students to select:
- We need to select a total of 3 students.
3. Combinations involving 2 boys and 1 girl:
- We need to find out how many ways we can select 2 boys from the 8 boys and 1 girl from the 4 girls.
- Ways to choose 2 boys out of 8:
- Use the combination formula [tex]\(\binom{n}{k}\)[/tex], where [tex]\(n\)[/tex] is the total number and [tex]\(k\)[/tex] is the number to choose. For boys, this becomes [tex]\(\binom{8}{2}\)[/tex].
- Ways to choose 1 girl out of 4:
- Similarly, use the combination formula. For girls, this is [tex]\(\binom{4}{1}\)[/tex].
4. Multiplying the results:
- The total number of ways to select 2 boys and 1 girl is the product of the two individual combinations.
Using this approach, we find that:
[tex]\[ \binom{8}{2} \times \binom{4}{1} = 28 \times 4 = 112 \][/tex]
So, there are 112 possible combinations to select 2 boys and 1 girl.
### (b) Selecting 3 boys only
1. Number of boys and girls:
- We still have 8 boys and 4 girls.
2. Combinations involving 3 boys:
- We need to find the number of ways to select 3 boys out of the 8 boys.
- Use the combination formula for this: [tex]\(\binom{8}{3}\)[/tex].
Using this approach, we find that:
[tex]\[ \binom{8}{3} = 56 \][/tex]
So, there are 56 possible combinations to select 3 boys only.
### Summary:
- (a) The number of ways to select 2 boys and 1 girl is 112.
- (b) The number of ways to select 3 boys only is 56.
