Question

7 different objects must be divided among 3 people .
In how many ways can this be done if one or two
of them must get no objects ?

Answer 1

\(\large \color{black}{\begin{align}
& \normalsize \text{7 different objects must be divided among 3 people .}\hspace{.33em}\\~\\
& \normalsize \text{In how many ways can this be done if one or two}\hspace{.33em}\\~\\
& \normalsize \text{of them must get no objects ?} \hspace{.33em}\\~\\
& a.)\ 381 \hspace{.33em}\\~\\
& b.)\ 36 \hspace{.33em}\\~\\
& c.)\ 84 \hspace{.33em}\\~\\
& d.)\ 180 \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

Is this a permutations/combinations question?

Answer 3

well when you do this, everyone gets 2 objects, theres one left, so there will be an uneven amount, what do you think it is

Answer 4

there is 3 people and seven objects, each one could get 2 and theres one left, there can also go 3 each, but that means 1 person will just get one object

Answer 5

Consider two cases :
1) exactly two people get 0 objects
2) exactly one person gets 0 object

Answer 6

How many ways are there for Case 1 ?

Answer 7

try again

Answer 8

3 way

Answer 9

Yes

Answer 10

a->0
b->0
c->7

a->7
b->0
c->0

a->0
b->7
c->0

total 3 way

Answer 11

2) exactly one person gets 0 object

Remove one person, and consider the \(7\) different objects.
Each object has \(2\) states: {person1, person2};
therefore, total possible ways for distributing these \(7\) different objects to the \(2\) people =
\(2*2*2*2*2*2*2 = 2^7\)

However, two of these are already counted in Case1, so total ways = \(2^7-2\)

Again, you could remove a person in \(3\) ways, so total ways of distributing \(7\) objects to \(3\) people such that exactly one of them gets \(0\) objects is \(3(2^7-2)\)

Answer 12

i still cant understand this,


\(2∗2∗2∗2∗2∗2∗2=2^7\)

Answer 13

Remove 3rd person, so now you have only two people : {person1, person2}

Answer 14

You want to distribute the 7 objects to those two people

Answer 15

Look at each object,
It can go to either `person1` or `person2`

so each object has `2` different states, yes ?

Answer 16

is this equivalent to find the number of whole number soln's

of a+b=7

Answer 17

yes it has 2 states

Answer 18

Nope, because the 7 objects in our case are not identical.

Answer 19

ok

Answer 20

Not just the number of objects, but also the type of object also matters

Answer 21

For example, below two are different ways :

way 1 :
Person1 : {o1}
Person2 : {o2, o3, o4, o5, o6, o7}

way 2 :
Person1 : {o2}
Person2 : {o1, o3, o4, o5, o6, o7}

Answer 22

looks like you need to add 3 to your solution

Answer 23

ok

Answer 24

is this also one of the way

way 1 :
Person1 : {o1,06}
Person2 : {o2, o3, o4, o5, o7}