Name: Claire
Student
Level: Secondary
Topic: Algebra and Number patterns
We need to be able to discover the nth term in a pattern of numbers and explain how we did it. We have arrived at the formula 1+2+3+...n=n(n+1)/2 for trios where a trio series is;
3 has the following trio (1,1,1)
4 has the following trios (2,1,1), (1,2,1), (1,1,2)
5 has the following trios (3,1,1), (1,3,1) (1,1,3) (1,2,2) (2,2,1) (2,1,2)
So the series is as follows
Term Value Trios
1 3 1
2 4 3
3 5 6
And so on.
Our problem is working out the formula for quads where a quad series is as follows
4 has the following quad (1,1,1,1)
5 has the following quads (1,1,1,2) (1,1,2,1) (1,2,1,1) (2,1,1,1)
etc etc
Hi Claire,
You need to know a bit about combinations, the number of ways of choosing k of n distinct objects, nCk.
The problems are equivalent to the following: Suppose that you have n identical lollipops and and m children come to your door; in how many ways can you hand out all of the lollipops such that each child gets at lest one lollipop? For example, with 5 lollipops and 3 children you can hand them out as (3,1,1), (1,3,1) (1,1,3) (1,2,2) (2,2,1) (2,1,2) where in the first case we have the first child gets 3 and the others get one each, and so on.
Suppose that you had 10 lollipops and 3 children, how can you count the ways? First, why not give each child one so that we don't have to worry about someone being left out. This leaves us with 7 lollipops to distribute to three children but we no longer have to worry about the someone getting none condition. Think of 7 x's and 2 |'s. For each ways of lining up these symbols we have a distribution of the lollipops - for example xx|xxxx|x means give the first child 2, the second 4 and the third 1; xxx||xxxx means give the 3, 0 and 4 respectively, and so on. Note in these two examples (because we already gave them 1 each) the children get (3,5,2) and (4,1,5). In how many ways can we line up these 9 symbols? If we choose 2 spots for the |'s then the x's must go n the other spots so we can do this in "9 choose 2" = 9C2 = 9(8)/2 ways. In general for n lollipops and 3 children you will get [(n-3)+2] choose 2 = (n-1)(n-2)/2 ways.
Suppose that you had 10 lollipops and 4 children, how
can you count the ways? First, why not give each child one so that
we don't have to worry about someone being left out. This leaves us
with 6 lollipops to distribute to 4 children but we no longer have
to worry about the someone getting none condition. Think of 6 x's and
3 |'s. In how many ways can we line up these 9 symbols? If we choose
3 spots for the |'s then the x's must go n the other spots so we can
do this in "9 choose 3" =
9C3 = 9(8)(7)/2 ways. As above, xx|x|x|xx gives us 2, 1,1,2 and adding
in the first lollipops that we gave out we have the solution (3,1,2,3),
etc. In general for n lollipops and 4 children you will get [(n-4)+3]
choose 3 = (n-1)(n-2)(n-3)/3! = (n-1)(n-2)(n-3)/6 ways.
Penny