Combinatorics

We are given that the polygon has 275 diagonals. The formula for the number of diagonals in a polygon with sides is: We are given that this is equal to 275: Multiply both sides by 2: Expanding the equation: Solve this quadratic equation using the quadratic formula: Thus, (since must be positive). Thus, the correct answer is 25.

We are given that the number of triangles that can be formed from sides is 220. Step 1: Use the combination formula The number of triangles that can be formed from vertices is given by: Step 2: Solve for We know that: Set this equal to 220: Multiply both sides by 6: Step 3: Solve the cubic equation Expanding the left side: By trial, we find . Thus, the correct answer is .

To solve this problem, we need to determine how many numbers greater than 1000 and less than 4000 can be formed using the digits 0, 2, 3, and 4, with repetition allowed. These numbers are four-digit numbers, and the thousands digit determines if the number is between 1000 and 4000.

For each specific thousands digit, the number of possible combinations for the hundreds, tens, and units digits is calculated as follows:

Now, compute the total number of numbers:

Adding these possibilities gives:

Therefore, the total number of numbers greater than 1000 but less than 4000 that can be formed is 128.

To determine the number of straight lines that can be drawn from 10 points where 4 are collinear, we can use the following approach:

So, the total number of straight lines:

Thus, the number of distinct lines that can be drawn is:

In conclusion, 40 straight lines can be drawn by joining any two of the 10 points in the plane.

The total number of questions is 12, divided into two groups, each containing 6 questions. We are required to choose 7 questions with the constraint that no more than 5 questions can be selected from either group. Step 1: Identify the possible combinations. Since we are selecting 7 questions, we have the following two possible scenarios for choosing questions from each group: - Choose 5 questions from one group and 2 questions from the other group. - Choose 4 questions from one group and 3 questions from the other group. Step 2: Calculate the number of ways for each scenario. # Scenario 1: Choose 5 questions from one group and 2 questions from the other group. - The number of ways to choose 5 questions from a group of 6 questions is given by . - The number of ways to choose 2 questions from the other group of 6 questions is given by . Thus, the number of ways for this scenario is: # Scenario 2: Choose 4 questions from one group and 3 questions from the other group. - The number of ways to choose 4 questions from a group of 6 questions is given by . - The number of ways to choose 3 questions from the other group of 6 questions is given by . Thus, the number of ways for this scenario is: Step 3: Total number of ways. Since there are two groups, we can have the following combinations: - Choose 5 questions from Group 1 and 2 questions from Group 2: - Choose 4 questions from Group 1 and 3 questions from Group 2: Thus, the total number of ways to select 7 questions is: Therefore, the correct answer is option (B)