Counting Principles

3. The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.

7. [latex]4+2=6[/latex]

9. [latex]5+4+7=16[/latex]

11. [latex]2\times 6=12[/latex]

13. [latex]{10}^{3}=1000[/latex]

15. [latex]P\left(5,2\right)=20[/latex]

17. [latex]P\left(3,3\right)=6[/latex]

21. [latex]C\left(12,4\right)=495[/latex]

23. [latex]C\left(7,6\right)=7[/latex]

31. [latex]\frac{8!}{3!}=6720[/latex]

33. [latex]\frac{12!}{3!2!3!4!}[/latex]

35. 9

37. Yes, for the trivial cases [latex]r=0[/latex] and [latex]r=1[/latex]. If [latex]r=0[/latex], then [latex]C\left(n,r\right)=P\left(n,r\right)=1\text{.\hspace{0.17em}}[/latex] If [latex]r=1[/latex], then [latex]r=1[/latex], [latex]C\left(n,r\right)=P\left(n,r\right)=n[/latex].

41. [latex]6 - 3+8 - 3=8[/latex]

43. [latex]4\times 2\times 5=40[/latex]

45. [latex]4\times 12\times 3=144[/latex]

47. [latex]P\left(15,9\right)=1,816,214,400[/latex]

49. [latex]C\left(10,3\right)\times C\left(6,5\right)\times C\left(5,2\right)=7,200[/latex]

51. [latex]{2}^{11}=2048[/latex]

53. [latex]\frac{20!}{6!6!8!}=116,396,280[/latex]

Binomial Theorem

1. A binomial coefficient is an alternative way of denoting the combination [latex]C\left(n,r\right)[/latex]. It is defined as [latex]\left(\begin{array}{c}n\\ r\end{array}\right)=C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}[/latex].

5. 15

9. 10

13. [latex]64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}[/latex]

15. [latex]27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}[/latex]

19. [latex]1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}[/latex]

21. [latex]\frac{1}{{x}^{4}}+\frac{8}{{x}^{3}y}+\frac{24}{{x}^{2}{y}^{2}}+\frac{32}{x{y}^{3}}+\frac{16}{{y}^{4}}[/latex]

23. [latex]{a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}[/latex]

25. [latex]{a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}[/latex]

27. [latex]3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}[/latex]

29. [latex]{x}^{24}-8{x}^{21}\sqrt{y}+28{x}^{18}y[/latex]

33. [latex]220,812,466,875,000{y}^{7}[/latex]

35. [latex]35{x}^{3}{y}^{4}[/latex]

39. [latex]\frac{1152{y}^{2}}{{x}^{7}}[/latex]

45. [latex]590,625{x}^{5}{y}^{2}[/latex]

49. The expression [latex]{\left({x}^{3}+2{y}^{2}-z\right)}^{5}[/latex] cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

Probability

7. [latex]\frac{1}{2}[/latex].

9. [latex]\frac{5}{8}[/latex].

11. [latex]\frac{1}{2}[/latex].

13. [latex]\frac{3}{8}[/latex].

14. [latex]S={HH,HT,TH,TT}[/latex].

15. [latex]\frac{1}{4}[/latex].

16. [latex]\frac{1}{2}[/latex].

17. [latex]\frac{3}{4}[/latex].

18. [latex]S={HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}[/latex].

19. [latex]\frac{3}{8}[/latex].

21. [latex]\frac{1}{8}[/latex].

23. [latex]\frac{15}{16}[/latex].

27. [latex]\frac{1}{13}[/latex].

29. [latex]\frac{1}{26}[/latex].

31. [latex]\frac{12}{13}[/latex].

33.

	1	2	3	4	5	6
1	(1, 1) 2	(1, 2) 3	(1, 3) 4	(1, 4) 5	(1, 5) 6	(1, 6) 7
2	(2, 1) 3	(2, 2) 4	(2, 3) 5	(2, 4) 6	(2, 5) 7	(2, 6) 8
3	(3, 1) 4	(3, 2) 5	(3, 3) 6	(3, 4) 7	(3, 5) 8	(3, 6) 9
4	(4, 1) 5	(4, 2) 6	(4, 3) 7	(4, 4) 8	(4, 5) 9	(4, 6) 10
5	(5, 1) 6	(5, 2) 7	(5, 3) 8	(5, 4) 9	(5, 5) 10	(5, 6) 11
6	(6, 1) 7	(6, 2) 8	(6, 3) 9	(6, 4) 10	(6, 5) 11	(6, 6) 12

35. [latex]\frac{5}{12}[/latex].

37. [latex]0[/latex].

39. [latex]\frac{4}{9}[/latex].

41. [latex]\frac{1}{4}[/latex].

43. [latex]\frac{3}{4}[/latex]

45. [latex]\frac{21}{26}[/latex]

47. [latex]\frac{C\left(12,5\right)}{C\left(48,5\right)}=\frac{1}{2162}[/latex]

49. [latex]\frac{C\left(12,3\right)C\left(36,2\right)}{C\left(48,5\right)}=\frac{175}{2162}[/latex]

56. [latex]\frac{40}{317}\times 100\%\approx 12.6\%[/latex]

57. [latex]\frac{C\left(40000000,1\right)C\left(277000000,4\right)}{C\left(317000000,5\right)}=36.78%[/latex]