Answer to Question #217553 in Discrete Mathematics for Bhatti

Question

 In a school, n+100 students have access to three software packages A, B andC           n-1 did not use any software  ,  n-2 used only packagesA                                    n-3 used only packages B     ,   n-4 used only packages C

 n-5 used all three packages    ,   n-6 used both A and B.

where n is 158

a)  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.

b) If twice as many students used package B as package A,write down a pair of   simultaneous equations in x and y. C)Solve the equations to find x and y. D) 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.