Calculation of Probability: At Least 3 Students from Group 1 in the Student Council Using Combinatorics

The problem of selecting six students from two groups to join the student council, where one group has 15 students and the other group has 10 students, provides a great opportunity to explore the application of combinatorics in probability calculations. In this article, we will step-by-step calculate the probability of having at least 3 students from group 1 in the student council.

Step 1: Total Number of Ways to Choose 6 Students

First, we calculate the total number of ways to select 6 students from a total of 25 students. This can be expressed as a binomial coefficient:

( binom{25}{6} frac{25 times 24 times 23 times 22 times 21 times 20}{6 times 5 times 4 times 3 times 2 times 1} 177100 )

Step 2: Calculate the Number of Ways for Different Scenarios

We consider scenarios where the number of students selected from group 1 is at least 3. These scenarios include selecting 3, 4, 5, and 6 students from group 1. For each scenario, we calculate the number of ways using binomial coefficients:

3 Students from Group 1 and 3 from Group 2

( binom{15}{3} frac{15 times 14 times 13}{3 times 2 times 1} 455 )

( binom{10}{3} frac{10 times 9 times 8}{3 times 2 times 1} 120 )

Total for this case: ( 455 times 120 54600 )

4 Students from Group 1 and 2 from Group 2

( binom{15}{4} frac{15 times 14 times 13 times 12}{4 times 3 times 2 times 1} 1365 )

( binom{10}{2} frac{10 times 9}{2 times 1} 45 )

Total for this case: ( 1365 times 45 61425 )

5 Students from Group 1 and 1 from Group 2

( binom{15}{5} frac{15 times 14 times 13 times 12 times 11}{5 times 4 times 3 times 2 times 1} 3003 )

( binom{10}{1} 10 )

Total for this case: ( 3003 times 10 30030 )

6 Students from Group 1 and 0 from Group 2

( binom{15}{6} frac{15 times 14 times 13 times 12 times 11 times 10}{6 times 5 times 4 times 3 times 2 times 1} 5005 )

Total for this case: ( 5005 times 1 5005 )

Step 3: Total Ways for At Least 3 Students from Group 1

Now we sum the totals from all cases:

Total for at least 3 from group 1: ( 54600 61425 30030 5005 151060 )

Step 4: Total Combinations

We already calculated the total combinations of selecting 6 students from 25:

( binom{25}{6} 177100 )

Step 5: Calculate the Probability

Finally, we can find the probability of selecting at least 3 students from group 1:

( P(text{at least 3 from group 1}) frac{151060}{177100} approx 0.852 )

or approximately 85.2%

Final Answer

Thus, the probability of having at least 3 students from group 1 in the student council is approximately 0.852 or 85.2%.