The image presents three math problems related to combinatorics. Problem 1: Two distinct dice are thrown. Find the number of possible outcomes where the sum of the numbers on the dice is either 5 or 6. Problem 2: There are 3 types of taxis available to travel to/from a village and a nearby town. When going to the town, a person can take any of the taxis or walk. However, on the way back, they decide to take a taxi. How many ways are there for them to travel to and from? Problem 3: There are 6 men and 5 women. They need to form mixed-gender pairs to play badminton. How many ways can these pairs be formed?

1. Problem Description

The image presents three math problems related to combinatorics.

Problem 1: Two distinct dice are thrown. Find the number of possible outcomes where the sum of the numbers on the dice is either 5 or

6.

Problem 2: There are 3 types of taxis available to travel to/from a village and a nearby town. When going to the town, a person can take any of the taxis or walk. However, on the way back, they decide to take a taxi. How many ways are there for them to travel to and from?

Problem 3: There are 6 men and 5 women. They need to form mixed-gender pairs to play badminton. How many ways can these pairs be formed?

2. Solution Steps

Problem 1:

Let's list the possibilities for each sum:

For a sum of 5: (1,4), (4,1), (2,3), (3,2). There are 4 possibilities.

For a sum of 6: (1,5), (5,1), (2,4), (4,2), (3,3). There are 5 possibilities.

Since the sum can be 5 or 6, we add the number of possibilities: 4 + 5 =

9.

Problem 2:

On the way to the town, the person can choose one of the 3 taxis or walk. So there are 3 + 1 = 4 options.

On the way back, the person can only choose a taxi. So there are 3 options.

The total number of ways is the product of the number of options for each trip: 4 * 3 =

1

2.

Problem 3:

We need to choose one man and one woman to form a pair.

There are 6 choices for the man.

There are 5 choices for the woman.

The total number of possible pairs is the product of these choices: 6 * 5 =

3

0.

3. Final Answer

Problem 1: 9

Problem 2: 12

Problem 3: 30