Is my thinking correct? combinations of 2 types, wee

In summary, the problem is asking for the number of 5 person teams that contain at most one man. The solution breaks this down into two steps - selecting teams without any men and selecting teams with one man. Using the addition rule and the multiplication rule, the final number of teams is determined to be 196. The process involves choosing men and women, with specific calculations for each step.
  • #1
mr_coffee
1,629
1
Hello everyone, another example in the book I'm wanting to make sure I'm seeing how they got their answer correctly.

It says:
Supppose the group pof 12 consists of 5 men and 7 women.
How many 5 person teams contain at most one man?

Solution:
The set of teams containing at most one man can be partitioned into the set that does not contain any men and the set that contains exactly one man. By the addition rule:

[number of teams with at most one man] = [# of teams without any men] + [# of teams with one man]
which equals:

5 choose 0 * 7 choose 5 + 5 choose 1 * 7 choose 4 = 21 + 175 = 196.



Okay I'm assuming they broke this down into a 2 step process.

#1. choose the men
#2. choose the women

So for the # of teams without any men
step 1:
You have 5 men to pick from, but you don't want to choose any, so you say 5 choose 0.

for step 2:
you want all women but no men, you have 7 women to choose from but you only want a group of 5, so the max you can choose from the 7 women is 5. So 7 choose 5, now apply the muliplcation rule as you get:

(5 choose 0) * (7 choose 5)

now for the
[# of teams with one man]
step 1:
choose men
step 2:
choose women

Step 1: Choose 1 man out of 5, 5 choose 1.
step 2: Choose 4 women out of the 7, so 7 choose 4, because you've already selected 1 man, so you have 4 slots open for the women.
(5 choose 1) * (7 choose 4)

Thanks!
 
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  • #2
Looks perfect.
 
  • #3
sweet thanks again
 

Related to Is my thinking correct? combinations of 2 types, wee

1. Is it possible to have different combinations of two types of thinking?

Yes, it is possible to have different combinations of two types of thinking. Our brains are complex and can often utilize multiple types of thinking simultaneously.

2. How can I determine if my thinking is correct?

The best way to determine if your thinking is correct is to gather evidence and evaluate it objectively. You can also seek feedback from others and consider alternative perspectives.

3. Can my thinking change over time?

Yes, our thinking can change over time as we acquire new knowledge and experiences. It is important to continuously evaluate and challenge our thinking to ensure it is accurate and up-to-date.

4. What are the two types of thinking?

The two types of thinking commonly referred to are analytical thinking and creative thinking. Analytical thinking involves breaking down complex problems into smaller parts and evaluating them logically. Creative thinking involves generating new and innovative ideas.

5. How can I improve my thinking skills?

There are many ways to improve your thinking skills, including practicing critical thinking, seeking out new experiences and perspectives, and continuously learning and challenging your beliefs. It is also important to maintain a healthy and open mindset.

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