Counting Ii Statistics Worksheet Page 3
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8
9Pascal’s Triangle
n
We can arrange the numbers
into a triangle.
k
0
0
1
1
0
1
2
2
2
0
1
2
3
3
3
3
0
1
2
3
4
4
4
4
4
0
1
2
3
4
5
5
5
5
5
5
0
1
2
3
4
5
6
6
6
6
6
6
6
0
1
2
3
4
5
6
7
7
7
7
7
7
7
7
0
1
2
3
4
5
6
7
n
In each row, the “top” number of
is the same. The “bottom” number
k
n
of
is the same in each upward slanting diagonal. The triangle continues
k
on forever. The first 8 rows are shown above.
This is called Pascal’s triangle. It is named after a French mathematician
who discovered it. It had been discovered outside of Europe centuries earlier
by Chinese mathematicians. Modern mathematics began in Europe, so its
traditions and stories tend to promote the exploits of Europeans over others.
n
Some values of
to start with
k
n
is the number of di↵erent ways you can select n objects from a set of n
n
objects. There is only one way to take everything – you just take everything
n
– so
= 1.
n
Similarly, there is only one way to take nothing from a set – just take
nothing, that’s your only option. The number of ways you can select nothing,
n
n
a.k.a. 0 objects, from a set is
. That means
= 1.
0
0
n
n
Now we can fill in the values for
and
into Pascal’s triangle.
n
0
43
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