Answer to Question #217560 in Discrete Mathematics for ismail khan

Question

In a school, 863+100 students have access to three software packages A, B and C.                         862 did not use any software. 861 used only packages A. 860 used only packages B. 859 used only packages C. 858 used all three packages. 857 used both A and B. (1) Draw a Venn diagram with all sets enumerated as for as possible. Label the two subsets which cannot be enumerated as x and y in any order. (2) If twice as many students used package B as package A, write down a pair Of simultaneous equations in x and y. (3) Solve the equations to find x and y. (4) How many students used package C?

Answer 1

a)

$$n-2+n-3+n-4+n-5+n-6+x+y=n+100$$

$$x+y=120-4n=>n\leq30$$

b)

$$n-3+n-5+n-6+y$$

$$=2(n-2+n-5+n-6+x)$$

$$3n-14+y=6n-26+2x$$

Then

$$\begin{matrix} x+y=120-4n\\ \\ 2 x-y=12-3n \end{matrix}$$

c)

$$\begin{matrix} 3x=132-7n\\ y=120-4n-x \end{matrix}$$

Let $n=12.$ Then

$$\begin{matrix} x=16\\ y=56 \end{matrix}$$

d)

$$8+7+16+56=87$$

87 students used package C.