Formula for no. of members in the sets

prachich1987 wrote:Hi,

I understand that the formula for calculating the total no of terms in two sets is as below

Total no of terms = no. of terms in set1 + no. of terms in set2 -Both+neither

I just wanted to ask whether we can use the same formula for calculating the no. of tgerms in three sets?

Thanks!!

For 3 sets,

Total number of terms =

terms in set 1 + terms in set 2 + terms in set 3 - terms common to set 1 & 2 but not 3 - terms common to set 2 & 3 but not 1 - terms common to set 3 & 1 but not 2 - 2*terms common to sets 1, 2 & 3 + terms not present in any of the 3 sets.

There are two versions of the formula, the one can be deduced easily using a venn diagram that is helpful :

1. Total = A+B+C - (AnB + BnC + CnA) + AnBnC + Neither
2. Total = A+B+C - [Sum of exactly two group members] - 2*AnBnC + Neither

Example #1:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections 3 an 4 on the diagram (assuming Marketing=A, Sales=B and Vision=C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections 2 and 4);
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.

Question: Total=?

Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members.

Total=M+S+V-(MnS+SnV+SnV)+MnSnV+Neither=20+30+40-(5+6+9)+4+0=74.

Answer: 74.


Example #2:
Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

Translating:
"Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs"
Total=59;
Neither=0 (as members are required to sign up for a minimum of one);
"22 students sign up for the poetry club": P=22;
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs": {Exactly 2 groups members}=6, so sum of sections 1, 2, and 3 is given to be 6, (among these 6 students there are no one who is the member of ALL 3 clubs)

"How many students sign up for all three clubs": question is PnHnW = x Or section 4 =?

Apply second formula:
Total = P+H+W - [Sum of exactly two group members] - 2*PnHnW + Neither
=> 59 = 22 + 27 + 28 - 6 - 2*x + 0
=> x = 6.

anshumishra wrote:There are two versions of the formula, the one can be deduced easily using a venn diagram that is helpful :

1. Total = A+B+C - (AnB + BnC + CnA) + AnBnC + Neither
2. Total = A+B+C - [Sum of exactly two group members] - 2*AnBnC + Neither

Example #1:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections 3 an 4 on the diagram (assuming Marketing=A, Sales=B and Vision=C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections 2 and 4);
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.

Question: Total=?

Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members.

Total=M+S+V-(MnS+SnV+SnV)+MnSnV+Neither=20+30+40-(5+6+9)+4+0=74.

Answer: 74.


Example #2:
Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

Translating:
"Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs"
Total=59;
Neither=0 (as members are required to sign up for a minimum of one);
"22 students sign up for the poetry club": P=22;
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs": {Exactly 2 groups members}=6, so sum of sections 1, 2, and 3 is given to be 6, (among these 6 students there are no one who is the member of ALL 3 clubs)

"How many students sign up for all three clubs": question is PnHnW = x Or section 4 =?

Apply second formula:
Total = P+H+W - [Sum of exactly two group members] - 2*PnHnW + Neither
=> 59 = 22 + 27 + 28 - 6 - 2*x + 0
=> x = 6.

Thanks for such a detailed explanation.
Can you plz advise on how to decide which formula to apply when...

prachich1987 wrote:

anshumishra wrote:There are two versions of the formula, the one can be deduced easily using a venn diagram that is helpful :

1. Total = A+B+C - (AnB + BnC + CnA) + AnBnC + Neither
2. Total = A+B+C - [Sum of exactly two group members] - 2*AnBnC + Neither

Example #1:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections 3 an 4 on the diagram (assuming Marketing=A, Sales=B and Vision=C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections 2 and 4);
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.

Question: Total=?

Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members.

Total=M+S+V-(MnS+SnV+SnV)+MnSnV+Neither=20+30+40-(5+6+9)+4+0=74.

Answer: 74.


Example #2:
Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

Translating:
"Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs"
Total=59;
Neither=0 (as members are required to sign up for a minimum of one);
"22 students sign up for the poetry club": P=22;
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs": {Exactly 2 groups members}=6, so sum of sections 1, 2, and 3 is given to be 6, (among these 6 students there are no one who is the member of ALL 3 clubs)

"How many students sign up for all three clubs": question is PnHnW = x Or section 4 =?

Apply second formula:
Total = P+H+W - [Sum of exactly two group members] - 2*PnHnW + Neither
=> 59 = 22 + 27 + 28 - 6 - 2*x + 0
=> x = 6.

Thanks for such a detailed explanation.
Can you plz advise on how to decide which formula to apply when...

You are welcome !
The first formula is good for the questions like example 1 and the 2nd one is good for the example type 2. That is why I have posted the two examples.