Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ objects from a set of $n$ distinct objects, without regard to order, is given by the binomial coefficient: ${}^n{C_r} = \frac{n!}{r!(n-r)!}$
• Problem Decomposition: Breaking a problem into mutually exclusive cases and summing the number of possibilities for each case.

Step-by-Step Solution

Step 1: Define the problem and possible cases

We need to form a team of 3 students from a group of 5 boys and $n$ girls, such that the team has at least one boy and at least one girl. This means we can have two possible cases: * Case 1: 1 boy and 2 girls * Case 2: 2 boys and 1 girl

Step 2: Calculate the number of ways for Case 1

We need to select 1 boy from 5 boys and 2 girls from $n$ girls. The number of ways to do this is given by: ${}^5{C_1} \cdot {}^n{C_2} = 5 \cdot \frac{n(n-1)}{2}$ Here, ${}^5{C_1}$ represents the number of ways to choose 1 boy from 5, and ${}^n{C_2}$ represents the number of ways to choose 2 girls from $n$.

Step 3: Calculate the number of ways for Case 2

We need to select 2 boys from 5 boys and 1 girl from $n$ girls. The number of ways to do this is given by: ${}^5{C_2} \cdot {}^n{C_1} = \frac{5 \cdot 4}{2 \cdot 1} \cdot n = 10n$ Here, ${}^5{C_2}$ represents the number of ways to choose 2 boys from 5, and ${}^n{C_1}$ represents the number of ways to choose 1 girl from $n$.

Step 4: Formulate the equation

The total number of ways to form the team is the sum of the number of ways for each case. We are given that this total is 1750. Therefore: ${}^5{C_1} \cdot {}^n{C_2} + {}^5{C_2} \cdot {}^n{C_1} = 1750$ Substituting the expressions we derived in Steps 2 and 3, we get: $5 \cdot \frac{n(n-1)}{2} + 10n = 1750$

Step 5: Simplify the equation

Divide both sides of the equation by 5: $\frac{n(n-1)}{2} + 2n = 350$ Multiply both sides by 2: $n(n-1) + 4n = 700$ Expand and simplify: $n^2 - n + 4n = 700$ $n^2 + 3n - 700 = 0$

Step 6: Solve the quadratic equation

We can factor the quadratic equation as follows: $(n - 25)(n + 28) = 0$ This gives us two possible solutions for $n$: $n = 25 \quad \text{or} \quad n = -28$

Step 7: Choose the valid solution

Since the number of girls, $n$, must be a positive integer, we discard the negative solution $n = -28$. Therefore, the only valid solution is $n = 25$.

Common Mistakes & Tips

• Forgetting to consider all cases: Make sure you've identified all the possible combinations of boys and girls that satisfy the "at least one boy and at least one girl" condition.
• Incorrectly calculating combinations: Double-check your calculations of ${}^n{C_r}$. It's easy to make a mistake with the factorials.
• Negative solutions: Remember that the number of people (girls in this case) cannot be negative. Always check for extraneous solutions.

Summary

We determined the number of ways to form a team with at least one boy and one girl by considering the two possible cases: one boy and two girls, or two boys and one girl. We calculated the number of ways for each case using combinations and then set up an equation based on the given total number of ways (1750). Solving the resulting quadratic equation gave us two possible values for $n$, but only the positive value, $n = 25$, is a valid solution.

Final Answer The final answer is \boxed{25}, which corresponds to option (B).