Permutations And Combinations

• Number of boys = 6
• Number of girls = 4
• Committee size = 5
• Condition: At least 2 girls must be included

Approach

We consider all possible cases that satisfy the condition of having at least 2 girls in the committee:

• 2 girls and 3 boys
• 3 girls and 2 boys
• 4 girls and 1 boy

We calculate each case separately and then sum the results.

Case 1: 2 Girls and 3 Boys

Case 2: 3 Girls and 2 Boys

Case 3: 4 Girls and 1 Boy

Total Number of Ways

Final Answer

The total number of ways to form the committee is 186.

Verification

We can verify by calculating the total possible committees without restrictions and subtracting the invalid cases:

This confirms our previous calculation.

We are given the function:
where denotes the permutation of objects taken at a time. The permutation function is defined as: for non-negative integers. Therefore, , and becomes:
Step 1: Understanding the Behavior of the Function
The factorial function grows very fast for increasing , but here will have a specific range depending on . We can calculate the values for small values of : For , is undefined. For , is undefined. For , is undefined. For , , so . For , , so . For , , so . For , , so .
Step 2: Conclusion from Calculation
After calculating several values for , it becomes evident that the function is constrained to certain values depending on . From the calculations and analysis, we conclude that the range of is .