Patterns and Secrets hidden in Pascal’s Triangle
Hemanthkumar B S
I B.Sc II Semester
U11GT21S0180
Govt. First Grade College, Tumkur
hemanthkumarbs03@gmail.com Mob: 7090519542
Introduction
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
This may look like a neatly arranged stack of numbers, but it’s actually a mathematical treasure trove. Indian mathematicians called it the Staircase of Mount Meru. In Iran, it’s the Khayyam Triangle. And in China, it’s Yang Hui's Triangle. To much of the Western world, it's known as Pascal’s Triangle after French mathematician Blaise Pascal.
The pattern that generates Pascal’s Triangle
Start with one and imagine invisible zeroes on either side of it. Add them together in pairs, and you’ll generate the next row. Now do that again and again.
0 + 1 + 0
0 + 1 + 1 + 0
0 + 1 + 2 + 1 + 0
0 + 1 + 3 + 3 + 1 + 0
Binomial Expansion
Each row corresponds to the coefficients of a binomial expansion of the form (x+y)2, where n is the number of the row.
|
Power |
Binomial Expansion |
Pascal’s Triangle |
|
0 |
(x+y)0=1 |
1 |
|
1 |
(x+y)1=1x+1y |
1,1 |
|
2 |
(x+y)2=1x2+2xy+1y2 |
1,2,1 |
|
3 |
(x+y)3=1x3+3x2y+3xy2+1y3 |
1,3,3,1 |
Powers of 2
Add up the numbers in each row, and you’ll get successive powers of 2.
|
Row Number |
Addition of Numbers |
In power of 2 |
|
0 |
1 |
20 |
|
1 |
1+1=2 |
21 |
|
2 |
1+2+1=4 |
22 |
|
3 |
1+3+3+1=8 |
23 |
|
4 |
1+4+6+4+1=16 |
24 |
Powers of 11
In each row, treat each number as part of a decimal expansion. It forms powers of 11 with respect to each row.
|
Row Number |
Decimal Expansion |
In power of 11 |
|
0 |
(1×1)=1 |
110 |
|
1 |
(1×10)+(1×1)=11 |
111 |
|
2 |
(1×100)+(2×20)+(1×1)=121 |
112 |
|
3 |
(1×1000)+(3×100)+(3×10)+(1×1)=1331 |
113 |
|
4 |
(1×10000)+(4×1000)+(6×100)+(4×10)+(1×1)=14641 |
114 |
Diagonals
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Probability
Pascal’s Triangle shows us the number of ways heads and tails can combine.
|
Number of Tosses |
Possible Results [Grouped] |
Pascal’s Triangle |
|
1 |
H T |
1,1 |
|
2 |
HH HT,HT TT |
1,2,1 |
|
3 |
HHH HHT,HTH,THH HTT,THT,TTH TTT |
1,3,3,1 |
Conclusion
The patterns in Pascal’s Triangle are a testament to the elegantly interwoven fabric of mathematics. And it’s still revealing fresh secrets to this day.