Pascal’s triangle gives a method for calculating the binomial coefficients: it begins as follows:
The nth row of this triangle gives the coefficients for
To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation
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Probability and Statistical Inference (9th Edition)
- Term of a Binomial Expansion Find the indicated terms in the expansion of the given binomial. 37. The 100th terms in the expansion of (1+y)100arrow_forwardConsider the expansion of (x+b)40. What is the exponent of b in the kth term?arrow_forwardDISCUSS • DISCOVER: Why ls (;) the Same as C(n,r)? This explains why binomial coefficients (;) that appear in expansion of (x + y)" are the same as C(n, r). num-bar Of ways of choosing r objects from n objects. first. Note that expanding a binomial using only the Distributive property gives (x+y)2=(x+y)(x+y) =(x+y)x+(x+y)y =xx+xy+yx+yy (x+y)3=(x+y)x+(xx+xy+yx+yy) =xxx+xxy+xyx+xyy+yxx +yxy+yyx+yyy (a) Expand (x+y)5 using only the Distributive Property. (b) Write all that the term represents x2y3 . These are all the terms that contain two x' s and three y's. (C) Note that tie two x's appear in all possible positions. Conclude that the number of terms that represent x2y3 is C (5,2) (d) In general, explain why (;) in the Binomial theorem is the same as C(n, r).arrow_forward
- Discovery and Writing If we applied the pattern of coefficients to the coefficient of the last term in Binomial Theorem, it would be n!n!(nn)!. Show that this expression equals 1.arrow_forwardUse the Binomial Theorem to write the first three terms of (2a+b)17.arrow_forwardStock is removed from a block in two operations. The original thickness of the block is represented by n. The thickness removed by the milling operation is represented by p and the thickness removed by the grinding operation is represented by t. What is the final thickness of the block?arrow_forward
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