Jimmi and Vishal are planning a dinner party. Vishal has 4 vegetarian friends and 3 non-vegetarian friends, while Jimmi has 3 vegetarian friends and 4 non-vegetarian friends. They decide to invite 4 friends in total and it was decided that both of them will invite two of their friends and also ensure that there are a total of 2 vegetarians and 2 non-vegetarians friends. How many ways can they invite their friends to the dinner party?

To determine the number of ways Jimmi and Vishal can invite their friends to the dinner party while meeting the given criteria, we can break down the problem step by step.

### Step 1: Determine Combinations for Vishal
Vishal needs to invite 1 vegetarian and 1 non-vegetarian friend out of his group of friends.

- **Number of ways to choose 1 vegetarian friend from 4:**
\[ \binom{4}{1} = 4 \]

- **Number of ways to choose 1 non-vegetarian friend from 3:**
\[ \binom{3}{1} = 3 \]

- **Total number of ways Vishal can invite his friends:**
\[ 4 \times 3 = 12 \]

### Step 2: Determine Combinations for Jimmi
Jimmi also needs to invite 1 vegetarian and 1 non-vegetarian friend out of his group of friends.

- **Number of ways to choose 1 vegetarian friend from 3:**
\[ \binom{3}{1} = 3 \]

- **Number of ways to choose 1 non-vegetarian friend from 4:**
\[ \binom{4}{1} = 4 \]

- **Total number of ways Jimmi can invite his friends:**
\[ 3 \times 4 = 12 \]

### Step 3: Calculate the Total Number of Combinations
To find the total number of ways they can invite their friends, we multiply the number of ways Vishal can choose his friends by the number of ways Jimmi can choose his friends.

\[ \text{Total combinations} = 12 \times 12 = 144 \]

### Conclusion
There are 144 ways Jimmi and Vishal can invite their friends to the dinner party while ensuring that there are 2 vegetarians and 2 non-vegetarians among the guests.

Would you like further details on any step, or do you have any other questions?

Here are 5 questions you can ask next:
1. How do you calculate combinations and permutations?
2. Can you explain binomial coefficients in more detail?
3. What if Vishal and Jimmi had different numbers of friends?
4. How would the solution change if they could invite 3 vegetarians and 1 non-vegetarian?
5. Can you solve a similar problem with a different number of guests?

**Tip:** When solving combinatorial problems, it's helpful to break down the problem into smaller parts and use basic combinatorial principles like combinations and permutations.

Learn how to calculate the number of ways Jimmi and Vishal can invite their friends to a dinner party while ensuring there are 2 vegetarians and 2 non-vegetarians among the guests. This problem involves combinatorial mathematics, focusing on binomial coefficients and combinations. Suitable for students in Grades 7-9.

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