Combinations

[fantasylo]

Dear all

I am trying to ask about the number of combinations for the following options

a
b
c1 / c2 / c3 (choose 1)
d1 / d2 / d3 (choose 1)
e1 / e2 (choose 1)

If I choose d1, I cannot choose d2 or d3.

For combination of 5, I will have 18 combinations.
a, b, c1, d1, e1
a, b, c1, d1, e2
a, b, c1, d2, e1
a, b, c1, d2, e2
a, b, c1, d3, e1
a, b, c1, d3, e2
a, b, c2, d1, e1
a, b, c2, d1, e2
a, b, c2, d2, e1
a, b, c2, d2, e2
a, b, c2, d3, e1
a, b, c2, d3, e2
a, b, c3, d1, e1
a, b, c3, d1, e2
a, b, c3, d2, e1
a, b, c3, d2, e2
a, b, c3, d3, e1
a, b, c3, d3, e2

Questions:
1) Is there a formula for this kind of scenario?

2) How to calculate the number of combinations for combination of 3?

Thank you for your assistance.

 

[Dr.Peterson]

Some more words would have helped clarify what you mean, but as I understand this, you have 1 choice for a, 1 choice for b, 3 choices for c, 3 choices for d, and 2 choices for e. Using the multiplication principle, there are a total of 1*1*3*3*2 = 18 ways to choose.

If you had to choose more than one from a category (e.g. 2 out of 3), then you would use the formula for combinations (also called binomial coefficients).

What have you learned about this topic?

 

[fantasylo]

Hi Peterson

I learned about combinations but this is not a straightforward combination. There are nested combinations within.

I have the following options

a
b
c1
c2
c3
d1
d2
d3
e1
e2

Limitations:
1) For c1/c2/c3, only 1 out of these is allowed (e.g. if c1 is chosen, then c2 and c3 cannot be chosen)
2) For d1/d2/d3, only 1 out of these is allowed
3) For e1/e2, only 1 out of these is allowed
4) There are 10 options, but the maximum I can choose is 5 options.

The question is how to determine the number of combinations for this kind of scenario?

1) Is there a formula for this kind of scenario?

2) How to calculate the number of combinations for combination of 3?

Good examples (should be counted):
a, b, c1
a, b, c2
a, b, c3

Bad examples (should not be counted):
a, c1, c2
c1, c2, c3

Hope it clarifies.

 

[Dr.Peterson]

I think what I said before still fits what you want, at least for this particular example. Why do you think my way of calculating 18 is incorrect?

 

[fantasylo]

I was asking whether there is a formula for this kind of scenario.

For combination of 5, the formula should be something like this:
2C2 * 3C1 * 3C1 * 2C1 = 18
(a,b) (c1/c2/c3) (d1/d2/d3) (e1/e2)

For combination of 3, I think the formula should be something like:

10C3 - (bad examples) = good examples

 

[Dr.Peterson]

Your first formula is equivalent to what I said.

But what is "the combination of 3"? You'll have to explain what you mean.

 

[Cubist]

I think I see what OP is trying to calculate. I would go through all the ways of choosing 3 from {a,b,c,d,e}, and then for each one calculate how many ways we can choose the "numeric suffix" of the letters (for c,d and e). So...

  abc  1*1*3 =  3 
  abd  1*1*3 =  3 
  abe  1*1*2 =  2 
  acd  1*3*3 =  9 
  ace  1*3*2 =  6 
  ade  1*3*2 =  6 
  bcd  1*3*3 =  9 
  bce  1*3*2 =  6 
  bde  1*3*2 =  6 
  cde  3*3*2 = 18

...and then sum these to obtain the result.

I can't think of a faster way of doing this.

 

[Dr.Peterson]

One way to do this, which is no faster by hand but is fast with technology, is to find the coefficient of x^2 in the expansion of (x+1)(x+1)(x+3)(x+3)(x+2), since that is formed by adding the products of 2 of the coefficients of x and 3 of the constants.

The answer is 68.

 

[fantasylo]

Dear all

Summary:

Cubist's solution:

a	b	c1/c2/c3		2C2	3C1			1 * 3 = 3​
a	b	d1/d2/d3		2C2	3C1			1 * 3 = 3​
a	b	e1/e2		2C2	2C1			1 * 2 = 2​
a/b	c1/c2/c3	d1/d2/d3		2C1	3C1	3C1		2 * 3 * 3 = 18​
a/b	c1/c2/c3	e1/e2		2C1	3C1	2C1		2 * 3 * 2 = 12​
a/b	d1/d2/d3	e1/e2		2C1	3C1	2C1		2 * 3 * 2 = 12​
c1/c2/c3	d1/d2/d3	e1/e2		3C1	3C1	2C1		3 * 3 * 2 = 18​
								68​

Dr.Peterson's solution:

x⁵ + 10x⁴ + 38x³ + 68x² + 57x + 18

My solution:

a/b/c1/c2/c3/d1/d2/d3/e1/e2				10C3			120​
Bad examples
a/b/d1/d2/d3/e1/e2	c1/c2/c3	c1/c2/c3		7C1	3C2		7 * 3 = 21​
a/b/c1/c2/c3/e1/e2	d1/d2/d3	d1/d2/d3		7C1	3C2		7 * 3 = 21​
a/b/c1/c2/c3/d1/d2/d3	e1/e2	e1/e2		8C1	2C2		8 * 1 = 8​
c1/c2/c3	c1/c2/c3	c1/c2/c3		3C3			1​
d1/d2/d3	d1/d2/d3	d1/d2/d3		3C3			1​
							52​
							120 - 52 = 68​

Thank you all for your assistance.