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Consider the recursive relationship for combinations:
C(n,r) = C(n-1,r) + C(n-1,r-1)
Prove this relationship algebraically using the mathematical definition of a combination, as well as that of the factorial function.
Provide a logical explanation for this relationship (Hint: consider n objects as consisting of n-1 existing objects plus a new nth object. Given the ultimate goal of selecting r objects from the entire set of n, you must select objects from the set of n-1 objects and decide how the new nth object should join the existing selected sets.)
Using graphing to check your answers is helpful. When you factor a trinomial into two binomials, each binomial represents a linear relationship. If you plot the two binomials (which are just lines) on a graph
Find the nodes x_i and the weights w_i so that the Gaussian quadrature of the sum from i=1 to 2 of (w_i) f(x_i) approximating the integral from -1 to 1 of f(x)dx is exact when f(x) is a polynomial of as high degree as possible.
A furniture manufacturer produces sofas, tables, and chairs. The profits per item are, respectively, $70, $120, and $80. The pieces of furniture require the following labor-hours for their manufacture.
Show that the functions x and x^2 are orthogonal in P5 with inner product defined by ( =sum from i=1 to n of p(xi)*q*(xi) ) where xi=(i-3)/2 for i=1,...,5.
A printer has a contract to print 100,000 invitations for a political candidate. He can run the invitations by using any number of metal printing plates from 1 to 20 on his press.
Assume f is continuous on the interval (a,b) that contains x0 and f(x0)>0. Show there is an open interval J that contains x0 and m>0 such that f(x)>=m for each x in J.
Find the probability that during any given day three machines will break down.
Find Probability of Drawing Cards. You draw two cards, which end up being Saddam Hussein and another one with his cousin Chemical Alia?.
Angular speed and linear speed - A bicycle wheel with a 26 inch radius is spinning at a rate of 4 revolutions per second.
The consumer price index for a new car in 1990 was 110.2, and in 1995 it was 136.5. If the price of the car was $12,880 in 1990, what was the price in 1995?
Finding the value of x such that the vertical distance is maximum - Determine the value of x for which d is maximum.
A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains.
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