Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

344 B. Selected Solutions
1.1 Exercises
1.1.1. There are 255 outfits. Use the multiplicative principle.
1.1.2.
(a) 8 ties. Use the additive principle.
(b) 15 ties. Use the multiplicative principle
(c) 5 · (4 + 3) + 7 42 outfits.
1.1.3.
(a) For example, 16 is the number of choices you have if you want to
watch one movie, either a comedy or horror flick.
(b) For example, 63 is the number of choices you have if you will watch
two movies, first a comedy and then a horror.
1.1.5.
(a) To maximize the number of elements in common between A and B,
make A ⊂ B. This would give |A ∩ B| 10.
(b) A and B might have no elements in common, giving |A ∩ B| 0.
(c) 15 ≤ |A ∪ B| ≤ 25. In fact, when |A ∩ B| 0 then |A ∪ B| 25 and
when |A ∩ B| 10 then |A ∪ B| 15.
1.1.6. |A ∪ B| + |A ∩ B| 13. Use PIE: we know |A ∪ B| 8 + 5 − |A ∩ B| .
1.1.7. 39 students. Use Venn diagram or PIE: 28+19+24−16−14−10+8 39.
1.1.8. 6 students don’t like potatoes.
1.1.9. 215 values of n.
1.1.12.
(a) 8 5 32768 words.
(b) 8 · 7 · 6 · 5 · 4 6720 words.
(c) 8 · 8 64 words.
(d) 64 + 64 − 0 128 words.
(e) (8 · 7 · 6 · 5 · 4) − 3 · (5 · 4) 6660 words.
1.2.1.
1.2 Exercises
(a) 2 6 64 subsets. We need to select yes/no for each of the six elements.
(b) 2 3 8 subsets. We need to select yes/no for each of the remaining
three elements.