<Question>
A Senate committee has 8 Republicans and 6 Democrats.  In how many ways can we form a subcommittee of 5 members that has at least one member from each party?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many ways we can form a subcommittee of 5 members from a Senate committee that has 8 Republicans and 6 Democrats. The key condition here is that the subcommittee must have at least one member from each party. Hmm, let's break this down step by step.

First, I remember that when dealing with combinations where order doesn't matter, like selecting committee members, we use combinations. The formula for combinations is C(n, k) = n! / (k! * (n - k)! ), where n is the total number of items, and k is the number we want to choose.

<prompt_1.1> Calculate the total number of ways to form a subcommittee of 5 members without any party restrictions using combinations. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So the total number of ways to choose 5 members out of 14 (since there are 8 + 6 = 14 senators) would be C(14, 5). Let me compute that: 14! / (5! * 9! ) = (14 × 13 × 12 × 11 × 10) / (5 × 4 × 3 × 2 × 1) = 2002. So there are 2002 total ways without any restrictions.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
But we need subcommittees with at least one Republican and one Democrat. That means we have to exclude the cases where all 5 are Republicans or all 5 are Democrats. So maybe the strategy here is to calculate the total number of possible subcommittees and then subtract the ones that don't meet the condition (i.e., all Republicans or all Democrats).

Let me check if that makes sense. So total subcommittees: C(14, 5) = 2002. Then subtract the all-Republican subcommittees and all-Democrat subcommittees.

<prompt_2.1> Calculate the number of all-Republican and all-Democrat subcommittees and subtract them from the total to find the valid subcommittees with at least one member from each party. </prompt_2.1>
-----

But let me verify this with another approach to make sure I didn't make a mistake. Another way is to consider that since we need at least one from each party, the possible distributions of party members in the subcommittee are:

<prompt_2.2> Explore an alternative method by considering the possible distributions of party members in the subcommittee and calculate the total using combinations. </prompt_2.2>
-----

Alternatively, I can think of it as the product of choosing 1 Republican and 1 Democrat, and then choosing the remaining 3 from the rest. But wait, that might complicate because the remaining members can be from either party. Let me try that approach.

<prompt_2.3> Consider another alternative method where the remaining members can be from either party and calculate the total using combinations. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Calculating all-Republican: There are 8 Republicans, so C(8, 5). Let's compute that: 8! / (5! * 3! ) = (8 × 7 × 6) / (3 × 2 × 1) = 56. 

All-Democrat: There are 6 Democrats, so C(6, 5) = 6. 

So the number of invalid subcommittees is 56 + 6 = 62. Therefore, the number of valid subcommittees is 2002 - 62 = 1940.
</execution_2.1>

<execution_2.2>
Republicans can be 1, 2, 3, or 4, and Democrats would be 4, 3, 2, or 1 respectively. Wait, but since the total is 5, the possible splits are (1,4), (2,3), (3,2), (4,1). So these are four different cases. If I calculate the combinations for each of these cases and sum them up, that should give the same result as the first method.
</execution_2.2>

<execution_2.3>
If we first choose 1 Republican and 1 Democrat, that's C(8,1)*C(6,1) = 8*6 = 48. Then, we need to choose 3 more members from the remaining 7 Republicans and 5 Democrats, which is 12 - 2 = 10 members? Wait, no, actually, after choosing 1 Republican and 1 Democrat, there are 7 Republicans left and 5 Democrats left, so total 12 members. So the number of ways to choose 3 from 12 is C(12,3) = 220. So then the total would be 48 * 220 = 10,560. Wait, that's way higher than before. That can't be right.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but hold on, let me confirm if this approach is correct. Sometimes when you subtract the invalid cases, you have to make sure you haven't made a mistake in calculating the invalid cases.

<prompt_3.1> Verify the calculation of invalid subcommittees and confirm the approach used to solve the problem. </prompt_3.1>
-----

Based on execution_2.2:
Let me compute each case:

<prompt_3.2> Calculate the number of valid subcommittees by considering the possible distributions of party members and summing up the combinations. </prompt_3.2>
-----

Based on execution_2.3:
Wait, that approach is wrong because when you fix 1 Republican and 1 Democrat, the remaining 3 can include any combination, but this method counts some cases multiple times. For example, a subcommittee with 2 Republicans and 2 Democrats would be counted multiple times depending on which two are initially chosen.

<prompt_3.3> Identify why the alternative method using combinations is incorrect and confirm the validity of the original approach. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
For all Republicans, we need to choose 5 out of 8. That's C(8,5) = 56, which seems right. All Democrats: 6 choose 5 is 6, correct. So 56 + 6 = 62 invalid. So 2002 - 62 = 1940.
</execution_3.1>

<execution_3.2>
For (1,4): C(8,1) * C(6,4). C(6,4) is the same as C(6,2) = 15. So 8 * 15 = 120. 

For (2,3): C(8,2) * C(6,3). C(8,2) is 28, C(6,3) is 20. So 28 * 20 = 560. 

For (3,2): C(8,3) * C(6,2). C(8,3) is 56, C(6,2) is 15. So 56 * 15 = 840. 

For (4,1): C(8,4) * C(6,1). C(8,4) is 70, C(6,1) is 6. So 70 * 6 = 420. 

Now, adding all these up: 120 + 560 + 840 + 420. Let's compute:

120 + 560 = 680

680 + 840 = 1520

1520 + 420 = 1940.
</execution_3.2>

<execution_3.3>
So that approach is not valid. Therefore, the original method of total minus invalid is correct. So the answer is 1940.

Alternatively, another way to think about it is using the principle of inclusion-exclusion. The total number is C(14,5). Then subtract the number of subcommittees that are all Republican and all Democrat. Since there are no overlapping cases here (a subcommittee can't be both all Republican and all Democrat), so inclusion-exclusion tells us that the formula is:

Total = All possible - (all Republican + all Democrat)
</execution_3.3>
</Execution_3>

<Plan_4>
Based on execution_3.2:
So both methods give the same result.
That’s reassuring. So the answer should be 1940. Let me just check once more for any possible mistakes.

<prompt_4.1> Verify the calculations for the alternative method and confirm the final answer. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
For the first method: 8 Republicans, 6 Democrats. Need 5 with at least one from each party. So subtract the all-R and all-D. C(14,5)=2002, C(8,5)=56, C(6,5)=6. 2002-56-6=1940. Correct.

Second method: sum over possible splits. Each split gives the same result. Calculations for each split:

1R 4D: 8 * 15 = 120

2R 3D: 28 * 20 = 560

3R 2D: 56 * 15 = 840

4R 1D: 70 * 6 = 420

Total: 120 + 560 = 680; 680 + 840 = 1520; 1520 + 420 = 1940. Correct.

So both methods confirm 1940. Therefore, I think that's the right answer. I don't see any mistakes in the reasoning. So the final answer should be 1940.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the number of ways to form a subcommittee of 5 members with at least one member from each party is $\boxed{1940}$.
</Final_answer>
</think>