To solve for the number of boys, [tex]\( n \)[/tex], given the total number of students is 25 and the probability that both students chosen are boys is [tex]\(\frac{7}{20}\)[/tex], follow these steps:
1. Understand the Given Information:
- Total students in the class: [tex]\( 25 \)[/tex]
- Probability that both students chosen are boys: [tex]\( \frac{7}{20} \)[/tex]
2. Express the Probability in Terms of Combinations:
- Probability that both students chosen are boys can be expressed using combinations:
[tex]\[
\frac{\binom{n}{2}}{\binom{25}{2}} = \frac{7}{20}
\][/tex]
- [tex]\(\binom{n}{2}\)[/tex] represents the number of ways to choose 2 boys from [tex]\( n \)[/tex], which is:
[tex]\[
\binom{n}{2} = \frac{n(n-1)}{2}
\][/tex]
- [tex]\(\binom{25}{2}\)[/tex] represents the number of ways to choose 2 students from 25, which is:
[tex]\[
\binom{25}{2} = \frac{25 \times 24}{2} = 300
\][/tex]
3. Set Up the Equation:
- Substituting the expressions for the combinations into the probability equation:
[tex]\[
\frac{\frac{n(n-1)}{2}}{300} = \frac{7}{20}
\][/tex]
- Simplify the equation:
[tex]\[
\frac{n(n-1)}{2} \times \frac{1}{300} = \frac{7}{20}
\][/tex]
[tex]\[
\frac{n(n-1)}{600} = \frac{7}{20}
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
- Cross-multiplying to get rid of the fraction:
[tex]\[
20n(n-1) = 7 \times 600
\][/tex]
[tex]\[
20n(n-1) = 4200
\][/tex]
- Divide both sides by 20:
[tex]\[
n(n-1) = 210
\][/tex]
5. Solve the Quadratic Equation:
- This simplifies to the quadratic equation:
[tex]\[
n^2 - n - 210 = 0
\][/tex]
- Solving this quadratic equation using the quadratic formula [tex]\( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -210 \)[/tex]:
[tex]\[
n = \frac{1 \pm \sqrt{1 + 840}}{2}
\][/tex]
[tex]\[
n = \frac{1 \pm \sqrt{841}}{2}
\][/tex]
[tex]\[
n = \frac{1 \pm 29}{2}
\][/tex]
- This gives two potential solutions for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{30}{2} = 15
\][/tex]
[tex]\[
n = \frac{-28}{2} = -14
\][/tex]
6. Interpret the Solution:
- Since the number of boys, [tex]\( n \)[/tex], cannot be negative, the only valid solution is:
[tex]\[
n = 15
\][/tex]
Thus, the number of boys in the class is [tex]\( 15 \)[/tex].
