*To*: mathgroup@smc.vnet.net*Subject*: [mg10916] Can anyone help me with these?*From*: "Liam McAllister" <lmcallister@email.msn.com>*Date*: Thu, 12 Feb 1998 20:15:58 -0500

1. The Binomial Coefficients n C r may be computed in Mathematica by typing Binomial[n,r]. Apart from the fact that nC0=nCn=1 for all positive integers n, there are equalities like 4C2=6C1=6 which make it rather difficult to work out how many different numbers occur as binaomial coefficients nCr as n ranges from 1 to N (say) and r ranges from 0 to n. Write a function h[] which will compute this number. Then use ListPlot[] (with PlotJoined->True) to compare N with h[N] as N ranges from 1 to 20. *************************************************** 2. Produce complete and accurate graphs of the functions defined by the formulae f(x)=cos(cos(2 Pi x)) and g(x)=cos(sin(2 Pi x)) on the interval [0,12] *************************************************** 3. Write a function d in Mathematica such that d(n) is the fraction of the first n Fibonnaci numbers which are prime. Find d(100). Use ListPlot[ ] with (PlotJoined->true) to give a graphical representation of the way d(n) varies with n for n =10, 20, ..., 100. Ensure that the viewer can see clearly where in the interval [0,1] d(n) lies. Fionnaci is f1=f2=1 and fn=fn-1 + fn-2 for n>2 Thanks in advance, Liam.