distribution & divided into sets

[sjpre10]

1. In how many ways can n things be shared between 2 people.

2. In how many ways can 9 books be distributed amongst a man, a woman and the child, if the man receives 4, the woman 3 and the child 2?

3. In how many ways can 8 boys be divided into two unequal sets?

4. In how many ways can 8 girls be divided into 4 sets of 2?

OA: 1: 2n-2, 2: 1260, 3: 92, 4: 105

How to solve these type of problems in combinations?
Please explain.

[thankont]

Q3) 8c1+8c2+8c3 = 8+28+56=92

[ionno]

For 2. I get 9*4*3*2*1 *3*2*1 - 9*2 = 1260

No idea what it means lol where's 12rk34 ?

[selytch]

Originally Posted by sjpre10

1. In how many ways can n things be shared between 2 people.

2. In how many ways can 9 books be distributed amongst a man, a woman and the child, if the man receives 4, the woman 3 and the child 2?

3. In how many ways can 8 boys be divided into two unequal sets?

4. In how many ways can 8 girls be divided into 4 sets of 2?

OA: 1: 2n-2, 2: 1260, 3: 92, 4: 105

How to solve these type of problems in combinations?
Please explain.

#1 the OA seems to be wrong... Also, not does having anything count as sharing? Is this a symmetric sharing?
IMO it is 2^(n-1)-1, assuming symmetry and having at least 1 item. (o/w 2^n)

#2 9C4*5C3*2C2 (126*10*1)=1260
we continue to choose from the same set, though its size is diminishing with each selection. You can actually choose any order 9C2*7C4*3C3 etc.

#3 agree with above (8c1+8c2+8c3 = 8+28+56=92), which is in how many ways one can select 1,2 and 3 boys out of 8 - with new set each time, hence addition.
#4 thinking...

Please stop child abuse dividing boys and girls!

[selytch]

Still not sure about #4...
Explanation #1 - imaging a binary line where each digit _position_ is the item and its _value_ is whether it belongs to person A or B (0 or 1).
The total number is 2^n. Since the distribution is symmetric (i.e. 010=101), the actual value is (2^n)/2=2^(n-1).
Sine we cannot have all 000 (or all 111 for that matter), subtract 1.

#2,#3 - fundamental principle of counting and addition.

#4 should be 8C2*6C2*4C2*2C2?