5.1
Let set A consists of an apple, an orange and a pear and set B consist of the apple and a banana.
list the elements of:
a) A and B
b) A or b
c) S
d) SnB
e) A U B U S
a) apple (intersection)
b) apple, banana, pear, orange
c) s= apple, banana, orange, pear
d) (Set of everything except set A)
= apple and banana
e) (everything) apple, banana, pear and orange
In how may ways can you choose 5 jellybeans from a package of 20 red jellybeans? Explain.
20C5 =15504 ways
At Subs Galore, you have a choice of lettuce, onions, tomatoes, green peppers, mushrooms, cheese, olives cucumbers and hot peppers to add to the sandwich. How many ways can you "dress" your sandwich?
There are 9 choices of topping, each one can be treated in 2 ways (you could either take it or leave).
Therefore, there are 29 = 512 possible "dressings" (including no toppings at all).
Rewrite this using Pascal's formula
19C5 + 19C6
Pascal's Formula
nCr =n-1Cr-1 + n-1Cr
Therefore,
=20C6
From the 220 graduating students in a school, 110 attended semi - formal and 150 attended the prom. If 58 students attended both events, how many students did not attend any event ?
n (AuB) = n(S) - n(P) - n(AnB)
= n(110) + n(150) - n(58)
= 202 - 220 students in total
= 18 did not attend any event.
In how many ways can a teacher select 5 students from a class of 23 to make a bulletin - board display?
23C5 = 33649 ways.
A school is trying to decide on new school colors. Students can choose 3 colors from gold, black, green, blue, red and white, but they kowtow another school has already chosen black, gold and red. How many different combinations of 3 colors can the students choose.
6C3 = 20
3C3 = 1 (since another school already chose those 3 colors).
20 - 1= 19
Describe how Pascal's triangle and Binomial theorem are related?
Pascal's triangle gives us the coefficients for an expanded binomial. Binomial theorem uses these coefficients to find the entire expanded binomial.
(a+b)n where n is the row number in Pascal's triangle
Would you classify Combination Lock, placed on the lockers, as a combination? why?
It'll be a permutation because order of arrangement matters in Permutation
If the lock as 23-35-42. It'll be just 1 number out of all the possible locker codes.
whereas, order of arrangements in a Combination doesn't matter.
23-35-42, 23-42-35,42-23-35,42-35-23,35-23-42,35-42-23 are all Combinations and can be arranged in different orders. They'll still be considered one option.
A survey of 1000 televisiom viewers conducted ny a local television station produced te followig data:
- 40% watch the news at 1200
- 60% watch the news at 1800
- 50% watch the news at 2300
- 25% watch the news at 1200 and 1800
- 20% watch the news at 1200 an 2300
- 20% watvh the news at 1800 and 2300
- 10% watch all 3 broadcasts
a) what percent of those surveyed at least one of these programs?
b) what percent watch non of these news broadcasts?
c) what percent view at 1200 and 1800, but not at 2300?
d) what percent view only one of these shows?
e) what percent view exactly 2 of these shows?
a) all of the percents add up to 95% (the views who watch at least one of these programs)
b)The remaining 5% do not watch any of these broadcast.
c) 50% of the viewers only watch news at 2300, so 45% must watch the news only at 1200 and 1800.
d) add up the percents outside the intersections to find the percent of viewers who watch only one of these shows: 5%+25%+20%= 50%
e) add up the percents in the intersections (except the one obtaining all 3) to find what percent view exactly 2 of these shows: 10%+10%=15% = 35%
In how many ways can a jury of 6 men and 6 women can be chosen from a group of 10 men and 15 women?
10C6 = 210 ways of selecting the men and 15C6 = 5005 ways of selecting the women.
so there are 210*5005 = 1051050 ways of selecting the jury.
A project team of 6 students is to be selected from a class of 30.
a) how many different teams can be selected?
b) Pierre, Gregory and Mark are students in the class, how many teams would include these 3?
c) How many teams would not include these 3?
a) 30c6 = 573775
b) If these 3 must be on a tea, then there are 27C3 = 2925 ways of selecting the rest of the team.
c) The number of teams without these 3 is given by 573775-2925= 590850.
Use Pascal's triangle to (i) predict the number of terms (ii) expand
[(2x+5y)2]2
(i) 4+1= 5 terms
(ii) (2x+5y)4
= (4C0)(2x)4(5y)0+(4C1)(2x)3(5y)1+(4C2)(2x)2(5y)2+(4C3)(2x)1(5y)3+(4C4)(2x)0(5y)4
= (1)(16x4)(1)+(4)(8x3)(5y)+(6)(4x2)(25y2)+(4)(2x)(125y3)+(1)(1)(625y4)
= 16x4+160x3y+600x2y2+100xy3+625y4
Suppose the Canadian Embassy in the Netherlands has 32 employers, who speak french and English. In addition to this, 22 of the employers speak German and 15 speak Dutch. If there are 10 who speak both German and Dutch, then how many of the employees speak neither German nor Dutch?
German: 22-10=12
Dutch: 15-10= 5
German nor Dutch: 32- 12- 10- 5= 5 speak neither.
There are 15 technicians and 11 chemists working in a research laboratory. In how many ways could they form a 5- member committee if
a) there are no restrictions
b) must have exactly 1 technician?
c) must have exactly 1 chemist?
d) must have exactly 3 technician ?
a) Total = 26
26C5= 665780 ways
b) 11C4 * 15C1
= 4950
c) 11C1 * 15C4
= 15015
d) 15C3 * 11C2 = 25025
There are 6 females and 5 males. You can only make a team of 7 members in it.
a) How many teams can you make with no restrictions?
b) How many teams can you make if there should be 3 males and 4 females in it?
a) 11C7= 330
b) 6C4= 15 for females
5C3= 10 for males
15 * 10= 150 teams with 3 males and 4 females in it
write 1024x10-3840x8+5670x6-4320x4+1620x2-243 in the form (a+b)n. Explain your steps
(a+b)n=(4x2-3)5
n= 10
1024x10 can be written as (4x2)5
(4x2)5= (4)5(x2)5= 1024x2*5= 1024x10
243 can be written as 3x=243 because it is divisible by 3
where x= log 243/ log 3= 5
35=243
Since the expansion has a pattern of - + - +..., we can conclude that the simplified equation would be (4x2-3)5
Jeffery works as a DJ at a local radio station. He chooses songs he will play based on requests which includes a list of 200 possible selections of:
- all the songs in the top 100
- 134 hard rock songs
- 50 phone in-requests
- 45 hard rock songs in the top 100
- 24 phone in-requests for hard rock songs
a) how many phone in-requests were for hard rock songs in the top 100?
b) how many of the songs in the top 100 were neither fine in-request nor hard rock selections?
a) let x represent the number in the intersection of all 3 sets. since there are a total of 200 selections, all of the segments add up to 200.
35+x+65+x+6+x+45-x+20-x+24-x+x = 200
x= 5
b) The number of songs in the top 100 which are neither hard-rock nor phone in-requests is given by 35+x= 40
35+5+40
A taxi is shuttling 11 students to a concert. The taxi can hold only 4 students. In how many ways can 4 students be chosen for?
a) the taxi's first trip?
b) The taxi's second trip?
a) There are 11C4 = 330 choices for the first trip
b) since 7 students are left now, there are 7C4 = 35 choices for the second trip.
a) How many different teams of 4 students could be chosen from the 15 students in Grade 12 Mathematics League?
b) How many of the possible teams would include the youngest student in the league?
c) How many of the possible teams would exclude the youngest student?
a) 15C4= 1365 teams
b)14C3= 364 teams where a spot is reserved for the youngest student
c) Teams excluding youngest student= Total teams- teams including the youngest student
= 1365-364
=1001 teams
Use your knowledge of algebra and the binomial theorem to expand and simplify
(25x2+30xy+9y2)3
=15625x6+56250x5y+84375x4y2+67500x3y3+30375x2y4+7920xy5+729y6
We know that (a+b)2= a2+2ab+b2
(5x+3y)2= 25x2+ 2(5x)(3y)+9y2
= 25x2+30xy+9y2
we sub in (5x+3y)2 in the equation
[(5x+3y)2]3 = (5x+3y)2*3=(5x+3y)6
Using Binomial theorem,
(5x+3y)6= (6C0)(5x)6(3y)0+(6C1)(5x)5(3y)1+(6C2)(5x)4(3y)2+(6C3)(5x)3(3y)3+(6C4)(5x)2(3y)4+(6C5)(5x)1(3y)5+(6C6)(5x)0(3y)6
= (1)(15625x6)(1)+(6)(3125x5)(3y1)+(15)(625x4)(9y2)+(20)(125x3)(27y3)+(15)(25x2)(81y4)+(6)(5x1)(243y5)+(1)(1)(729y6)
=15625x6+56250x5y+84375x4y2+67500x3y3+30375x2y4+7920xy5+729y6