The Chemistry Maths Book, Second Edition

200 Chapter 7Sequences and series

(ii) By equation (7.14),

0 Exercises 26–33

The binomial coefficients form a pattern of numbers called the Pascal triangle

5

1

11

121

1331

14641

15101051

1 =

Each row begins and ends with the number 1 for the coefficients of x

n

and y

n

in

the expansion of(x 1 + 1 y)

n

, and each interior number is the sum of the two numbers

diagonally above it.

The multinomial expansion

This is the generalization of the binomial expansion (7.14),

(7.15)

in which the k-fold sum is over all positive integer and zero values ofn

1

, n

2

,1=1, n

k

,

subject to the constraintn

1

1 + 1 n

2

1 +1-1+ 1 n

k

1 = 1 n. The multinomial coefficients

n

nn n

n

nn n

k

k

12

12





















=

!

!!!

()

!

!!!

xx x

n

nn n

xx x

k

n

k

k

nn n

n

k

k

12

12

12

12

+++ =

∑







nnn

1 2

∑∑

=+ +243 405 270 90 15+ + +

2345

xxxxx

()xxx+=

















• 
  

















• 
  









3

5

0

3

5

1

3

5

2

50514









• 
  

















• 
  

















• 
  



xxx

23 32 41

3

5

3

3

5

4

3

5

 5













x

50

3

5

Blaise Pascal (1623 –1662). French philosopher and mathematician who made contributions to geometry, the

calculus and, with Fermat, developed the mathematical theory of probability (at the instigation of the gambler

Antoine Gombard, Chevalier de Méré). The Pascal triangle appears in the Traité du triangle arithmétique,

avec quelques autres petits traités sur la même manière, published posthumously in 1665. The work contains a

discussion of the properties of the binomial coefficients, with applications in games of chance. The triangle was

known long before; it appeared in a book by the Chinese mathematician Yang Hui in 1261, and the properties of

the binomial coefficients were discussed by the Persian Jamshid Al-Kashi in his Key to arithmetic, (c. 1425).