Discrete Maths Question Using Inclusion Exclusion: A gardener buys 10 rosebushes....

[Mathsgirl2]

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red,
pink, white, and yellow bushes (indistinguishable apart from colour). How
many different collections of bushes can she buy
(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

 

[Deleted member 4993]

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red,
pink, white, and yellow bushes (indistinguishable apart from colour). How
many different collections of bushes can she buy
(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[stapel]

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red, pink, white, and yellow bushes (indistinguishable apart from colour). How many different collections of bushes can she buy:

(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

I'm guessing you're working on Exercise 2(ii) on this exam paper. (Your spelling and line breaks are exactly the same, suggesting a copy-and-paste.) Did you have any trouble with the first part of the exercise? That part stated:

---

2. (i) (a) State the inclusion-exclusion formula. Explain clearly the meaning of any special notation that you use.

Consider 6-letter words constructed using the letters X, Y, and Z.

(b) How many such words are there in total?
(c) How many contain at least one X and at least two Ys?

---

How far have you gotten in applying the above information to part (ii) of the exercise (that is, to the part that you posted)?

Please be complete. Thank you!

 

[Mathsgirl2]

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

For the first part I did 4^10=1048576
For the second part I tried to consider |Ai| to be one of the colours being an even number of times or not at all: 3^10+3^8+3^6... =66430
but the I'm not sure how I'd find |Ai n Aj| ec.

 

[pka]

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red,
pink, white, and yellow bushes (indistinguishable apart from colour). How
many different collections of bushes can she buy
(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

You must show your work or tell us what about this question stands in your way.
In this case, I suggest it is the latter that is the case.
There a simple approach to part (a), look into multi-selections.
However parts (b) & (c) are nightmares if you do not know how to use generating functions.
On the other hand, if you know how then look at this. The answer to part (c) is there.

 

[pka]

For the first part I did 4^10=1048576
For the second part I tried to consider |Ai| to be one of the colours being an even number of times or not at all: 3^10+3^8+3^6... =66430 but the I'm not sure how I'd find |Ai n Aj| ec.

I am sorry that I did not see your above reply before I posted.
Sorry to tell you that none of those answers is correct.

The answer of (a) is \(\displaystyle \dbinom{10+4-1}{10}=286\) That is a well known multi-selection formula.
Also see this. In that expansion look at the coefficient of \(\displaystyle x^{10}\).

For part (c), look at this

 

[Ishuda]

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red,
pink, white, and yellow bushes (indistinguishable apart from colour). How
many different collections of bushes can she buy
(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

Something is missing. for example, let the number of rose bushes available be
red = 10 n
pink = n
white = 5 n
yellow = 2 n
and then let n increase without bounds so the number of each color of bud is unlimited.

You would get a totally different answer than if it were
red = 1000000 n
pink = n
white = n
yellow = n
where n increases without bounds.

 

[pka]

Something is missing. for example, let the number of rose bushes available be
red = 10 n,_____pink = n,_____white = 5 n,_____yellow = 2 n,_____
and then let n increase without bounds so the number of each color of bud is unlimited.
You would get a totally different answer than if it were
red = 1000000 n,_____pink = n
white = n,_____,_____yellow = n
where n increases without bounds.

I think that you have really missread this question.

A gardener buys 10 rosebushes from a shop which has unlimited stocks of red,
pink, white, and yellow bushes (indistinguishable apart from colour). How
many different collections of bushes can she buy
(a) in total?
(b) in which every colour is either used an even number of times or not at all?
(c) in which every colour is either used an odd number of times or not at all?

Even if the supplier had only forty bushes in stock, ten of each of four colours, it would not change the answers. The gardener is buying only ten plants.

 

[Ishuda]

I think that you have really missread this question.

Even if the supplier had only forty bushes in stock, ten of each of four colours, it would not change the answers. The gardener is buying only ten plants.

Yes, you're correct. I was thinking there was a 'probability question in there' in which case the ratio of one color to another would have some bearing on what was chosen 'at random'.