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Identities on the Binomial Coefficients

Identities on the Binomial Coefficients. There are a great number of patterns in the Pascal triangle. Some are quite straightforward to recognise and to prove, others are more complicated.

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Identities on the Binomial Coefficients

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  1. Identities on the Binomial Coefficients • There are a great number of patterns in the Pascal triangle. Some are quite straightforward to recognise and to prove, others are more complicated. • They can be very important in any application of the binomial theorem, and many of them will reappear in the chapter on probability. • Each pattern in the Pascal triangle is described by an identity on the binomial coefficients nCk— these identities have a rather forbidding appearance, and it is important to take the time to interpret each identity as some sort of pattern in the Pascal triangle. • Three approaches to generating identities from this expansion are developed in turn: • substitutions, • methods from calculus, differentiation and integration, • and equating coefficients.

  2. Example 1 Find • The term will come from the coefficient of is + +

  3. Method 1 - Substitution Substitutions into the basic binomial expansion or any subsequent development from it will yield identities. We begin with the expansion of Example 2 Prove the sum of coefficients in (a + b)nis equal to 2n. (a + b)n= Let a = (1+ 1)n=  OR

  4. Method 2 - Differentiation The basic expansion can be differentiated before substitutions are made. In these questions the power is normally Example 3 (1 + x)2n= Differentiating both sides 2n(1 + x)2n-1= Let a = 2n(1 )2n-1= 0=

  5. Method 3 - Integration The basic expansion can be integrated before substitutions are made. As always, integration involves finding an unknown constant. In these questions the power is normally Example 4 (1 + x)n= Integrating both sides Instead of having a constant on each side, just use K = Let x = 0 =

  6. Example 4 = Let x = = = =

  7. Method 4 - Equating coefficients This involves taking two equal expansions and equating coefficients. Example 5 B prove multiplied by or last term, ,

  8. Example 5 B prove that

  9. Today’s work Exercise 10.5 Page 492→493 Q1b+c Q2→5, 7 Q8a, 9, 11→15 Yesterday’s work Exercise 10.4 Page 488 → 489 Q1 → 3 Q7, 9, 11 & 17 Exercise 10.4 Page 488 → 489 Q18 → 21

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