•
•
•
•
It’s a triangle.
A triangle of numbers!
Pascal did not create it…. The Chinese did.
Blaise Pascal discovered all of the unique
patterns in it.
1
1
1
4
1
3
1
2
6
1
3
1
4
1
1
Then we add the left
Continue
and right
numberwith this addition for each line
together on the
second row
First we start off with a triangle of ones
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
18
19
20
190
20
56
120
35
126
1
5
15
70
210
1
4
10
35
84
816
3060
56
252
1
6
21
126
4845
8568
1
7
28
84
210
15504
18564
1
8
36
120
38760
31824
1
9
45
77520
43758
1
10
125970
48400
1
167740
42438
182996
161800
28524
340
5268
26440
120
1740
1
16
136
596
2336
9344
1
15
460
7008
1
14
105
1280
19432
62120
91
940
14164
1
13
235
3988
42688
113650
144
3048
1
12
78
705
10176
70962
66
561
7128
11
66
2343
18348
90838
0
1782
11220
55
495
4785
24090
92158
495
3003
12870
165
1287
6435
24310
75582
792
3432
11440
330
1716
6435
19448
50388
924
3003
8008
462
1716
5005
12376
27132
792
2002
4368
462
1287
3003
6188
11628
495
1001
1820
330
715
1365
2380
3876
220
364
560
165
286
455
680
969
66
91
120
153
1140
45
55
78
105
136
171
12
14
16
11
13
15
17
15
28
36
6
10
21
1
3
Just imagine 40 rows of a Triangle!
1
1
9
10
5
7
8
3
4
6
1
2
153
749
3085
1
17
1
18
171
920
1
19
190
1
20
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
The very top
is Row 0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
12
13
14
9
66
220
286
364
495
2002
21
792
36
495
9
1
10
55
0
495
1782
1
45
165
1287
3003
1
8
120
330
1716
3432
7
84
462
1
28
210
924
1
6
56
126
1716
3003
15
252
792
1
5
35
462
1287
4
70
126
330
715
1001
35
210
1
10
20
56
84
165
6
15
120
1
3
10
21
36
2
4
28
45
55
78
91
7
1
3
5
6
8
10
11
Each row has
a reference
number
1
11
66
66
561
1
1
12
78
144
1
13
91
What
The
sum
is the
of all
sum
theofnumbers
the eighth
in row?
a row = 2Row Number
The answer
sum of row
is 286or= 256
26 or 64
1
14
1
Theofnext
All
first
these
element
number
1’s areis
in each
always
element
row
would
element
0 be element
zero 1
1
1
1
2
1
1
1
1
1
1
1
1
1
12
13
14
9
55
66
78
91
220
330
495
1001
1287
2002
924
3003
84
792
1
36
9
45
165
495
1287
3003
1
8
120
330
1716
3432
7
28
210
462
1716
1
21
126
462
1
6
56
252
792
715
15
35
126
1
5
70
210
165
286
364
84
120
4
10
35
56
36
1
20
21
28
45
10
15
7
8
10
11
6
1
3
6
5
1
1
3
4
1
1
55
0
495
1782
1
10
1
11
66
66
561
1
12
78
144
1
13
91
1
14
th row!
Each numberLet’s
or element
in a6row
has a reference
look at the
number starting with the number 1.
1
Element 0
Element 1
6
Element 3
15 20 15
Element 2
Element 4
Element 5
6
1
Element 6
1
1
Now let’s
We’re
at the
go to
6ththe
row
3rd
element
0
1
1
1
1
1
1
1
1
1
12
14
66
165
220
286
364
495
2002
21
792
36
495
9
1
10
55
0
495
1782
1
45
165
1287
3003
1
8
120
330
1716
3432
7
84
462
1
28
210
924
1
6
56
126
1716
3003
15
252
792
1
5
35
462
1287
4
70
126
330
715
1001
35
210
1
10
20
56
84
1
3
6
15
120
3
10
21
36
1
2
3
4
28
45
55
78
91
7
9
2
5
6
8
10
11
13
1
1
1
1
1
1
11
66
66
561
1
1
12
78
144
Let’s find the 3rd element in 6th row
1
13
91
1
14
1
1
3rd
Here is the
element in 6th row
1
1
1
1
1
1
1
1
1
7
1
3
3
6
15
10
20
84
35
70
1
6
21
126
252
1
7
56
126
210
1
5
15
35
56
1
4
10
21
120
3
4
28
36
45
2
2
5
6
8
9
10
1
1
1
1
28
84
8
36
210
120
1
9
45
1
10
1
11
55
165
330
462
462
330
165
55
11
1
“!” 1is a1 factorial.
Start
with
the
5!
number
=
5×4×3×2×1
and
multiply
or
120
by
every
sequential
number
12
66
220
495
792
924
792
495
0
66
12
1
10! =715
10×9×8×7×6×5×4×3×2×1
down
to
or 3,628,800
1
13
78
286
1287
1716
17161
1287
495
66
78
13
1
1
14
91
364
1001
2002
3003
3432
3003
1782
561
144
91
14
1
Find 6C3 (nCr) or the 6th row choose 3rd element
n!
_______
r!(n-r)!
6×5×4×3×2×1
_______
3×2×1(6-3)!
720
_____
6(3)!
720
_____
6(3×2×1)
720
_____
= 20
36
• Let’s find the 5 element in the 15th row
• We are finding nCr or 15C5.
• We are using our formula with n being the row and r being
the element.
nCr =
n!
_______
r!(n-r)!
5C15 =
15!
_______
5!(15-5)!
1307674368000
_______
120(3628800)
1
Go toover
Now
the 15
tothwhere
row
5th
the
element
would be
1
1
1
1
1
1
1
1
1
14
9
11
12
13
66
220
286
364
495
2002
21
792
36
495
9
1
10
55
0
495
1782
1
45
165
1287
3003
1
8
120
330
1716
3432
7
84
462
1
28
210
924
1
6
56
126
1716
3003
15
252
792
1
5
35
462
1287
4
70
126
330
715
1001
35
210
1
10
20
56
84
165
6
15
120
1
3
10
21
36
2
4
28
45
55
78
91
7
1
3
5
6
8
10
1
1
1
1
1
11
66
66
561
1
1
12
78
144
1
13
91
1
14
3003
Add together the two number above the 5th spot.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
14
9
10
11
12
13
66
220
364
495
2002
21
792
36
495
9
1
10
55
0
495
1782
1
45
165
1287
3003
1
8
120
330
1716
3432
7
84
462
1
28
210
924
1
6
56
126
1716
3003
15
252
792
1
5
35
462
1287
4
70
126
330
715
1001
35
210
1
10
20
56
84
165
6
15
120
1
3
10
21
36
286
4
28
45
55
78
91
7
8
2
3
5
6
1
11
66
66
561
1
1
12
78
144
1
13
91
1
14
1