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PASCAL'S TRIANGLE

We will see in this chapter how Fibonacci numbers can be extracted from Pascal's triangle. In addition, we will derive a host of Fibonacci and Lucas identities involving binomial coefficients.

We will begin with a brief discussion of binomial coefficients, which are coefficients occurring in the binomial expansion of c011-math-001. The term binomial coefficient was introduced by the greatest German algebraist Michel Stifel (1486–1567).

The earliest known occurrence of binomial coefficients can be traced back to a tenth-century commentary by the Indian mathematician Halayudha on Pingala's Chandas Shastra; interestingly, it also contains a discourse on what is now called Pascal's triangle (called Meru Prastara). Bhaskara (1114–1185?) gives a concise discussion of binomial coefficients in his 1150 classic work Leelavati.

11.1 BINOMIAL COEFFICIENTS

Let n and k be nonnegative integers. The binomial coefficient c011-math-002 is defined by

equation
Pascal*A painting of Pascal.

It is also denoted by and . Read as “n choose ...

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