In how many ways can 7 people be divided into 4 groups ?

[paul_boz]

I think the answer is 875 but I'm far from confident.

There are only 3 ways to make up 4 teams from 7 people;
A. 1,1,1,4
B. 1,1,2,3
C. 1,2,2,2

A gives 35 combinations
B gives 210 combinations
C gives 630 combinations

So the total is 875. Am I right or wrong ?

Many thanks in advance.

[Guest]

In my opinion, you are right.

[paul_boz]

Actually, it looks like I'm wrong as the answer to C is in fact 630/3!=105 giving a grand total 350.

See [url]https://math.stackexchange.com/questions/1950304/combinatorics-dividing-people-into-groups
[/url]

[Guest]

No, 875 is not correct. Your method of breaking the problem into the possible team-size patterns is the right approach, but the counting in cases B and C needs correction.

The possible distributions are:

**A) 4,1,1,1**
Choose the group of 4 people:

C(7,4) = 35

**B) 3,2,1,1**
Choose the group of 3, then the group of 2, and the remaining two people form the single-person teams:

C(7,3) × C(4,2) × C(2,1) ÷ 2! = 210

(The division by 2! avoids counting the two identical single-person teams twice.)

**C) 2,2,2,1**
Choose the single person, then divide the remaining 6 people into three pairs:

7 × (6! ÷ (2!² × 3!)) = 105

Therefore, the total number of ways is:

35 + 210 + 105 = **350**

So the correct answer is **350**, not 875.

You can also check calculations with my tool: [Long Div Calc]https://longdivcalc.com/

[nycmath]

I think the answer is 875 but I'm far from confident.

There are only 3 ways to make up 4 teams from 7 people;
A. 1,1,1,4
B. 1,1,2,3
C. 1,2,2,2

A gives 35 combinations
B gives 210 combinations
C gives 630 combinations

So the total is 875. Am I right or wrong ?

Many thanks in advance.

See attachment.

[Guest]

You’re very close, but there’s one more case to include. For C, the number of combinations is **7 × 15 = 105**, since you choose the single person in 7 ways and then split the remaining 6 people into 3 pairs in 15 ways.

So the total is:

**35 + 210 + 105 = 350 combinations.**

Therefore, I believe the answer is **350, not 875**.

If you also need to convert the resulting data from PDF to Excel, this tool may be useful: https://convert-online.net/pdf-to-excel/

[nycmath]

You’re very close, but there’s one more case to include. For C, the number of combinations is **7 × 15 = 105**, since you choose the single person in 7 ways and then split the remaining 6 people into 3 pairs in 15 ways.

So the total is:

**35 + 210 + 105 = 350 combinations.**

Therefore, I believe the answer is **350, not 875**.

If you also need to convert the resulting data from PDF to Excel, this tool may be useful: https://convert-online.net/pdf-to-excel/

You take it from here.