a. A president, vice president, and secretary are chosen from a twenty-five member garden club. Each of the three has a distinct role. As a result, this involves permutations. The total number of ways in which this can be done is P(25, 3) = 13800.
b. A cook chooses 5 potatoes from a bag of 12 potatoes to make a potato salad. Here, the 5 potatoes being chosen are not distinct. Therefore, this involves a combination. The total number of ways of choosing 5 potatoes is C(12, 5) = 792.
c. A teacher makes a seating chart for 22 students in a classroom of 30 desks. Each of the students in the class is distinct. The number of ways of making the chart is given by a permutation. This is equal to P(30, 22) = 6.57*10^27.
2) There are 680 three-digit numbers that are available for use as area codes in North America. As of April 2010, 301 of them were actually being used.
a. The number of additional three-digit codes available for use is the difference of 680 and 301, which is equal to 379.
b. If a telephone number within a given area code has n digits, each of the digits can have 10 theoretical values. This gives a total number of possible numbers as 10^n.
c. The total numbers possible in North America is equal to the product of area codes in use and the number of theoretical values under each area code. This is 301*10^n.
3) The word "logarithm" has 9 distinct letters. The number of 9-letter words that can be formed is P(9, 9) = 362880.
4)The professor selects 8 questions from the 20 that have been given. The order in which they are chosen is not relevant here. Therefore the questions can be selected in C(20, 8) = 125970.
5)The coefficient of x^p*y^(n-p) in the binomial expansion of (x+y)^n is C(n, p). In the question, n is equal to 14 and p is equal to 11. The coefficient of this term is C(14, 11) = 364.
6) The Binomial Theorem gives (x + y)^n = `sum_(p=0)^n C(n,p)*x^n*y^(n-p)` ` `
` `(2x-3y)^5 = C(5,0)*(2x)^0*(-3y)^5+C(5,1)*(2x)^1*(-3y)^4+C(5,2)*(2x)^2*(-3y)^3+C(5,3)*(2x)^3*(-3y)^2+C(5,4)*(2x)^4*(-3y)^1+C(5,5)*(2x)^5*(-3y)^0
7)Sal opens a box of a dozen chocolate cremes and generously offers two of them to his friend Val. Val likes vanilla cremes the best, but all of the chocolates look alike on the outside. Four of the cremes are vanilla.
The probability that both of Val's picks are vanilla is (4/12)*(3/11) = 12/132 = 1/11
8) Michael makes 90% of his free throws in basketball. He shoots 20 free throws, and his chance of making each one is independent of the other shots. The probability that he makes:
a. all 20 : (.9)^20
b. exactly 18 : (.9)^18*(.1)^2
c. at least 18: (.9)^18*.1^2 + (.9)^19*.1 + (.9)^20*.1^2
9)In the game Yahtzee, on the first roll five dice are tossed simultaneously. The probability of rolling five of a kind is equal to 5*(1/6)^4.
Let's say we are first choosing a president, then vice-president and then secretary. We can choose president in 25 ways (1 out of 25 members). Now we have only 24 members to choose from so we can choose a vice-president in 24 ways and analogously we can choose secretary in 23 ways. Hence 25*24*23=13800. If we were choosing 3 members for the same function then we would have combinations because there is no way to differentiate them.
Because potatoes are all the same (mathematically speaking) so their order is not important. `((12),(5))=792`
First student can be seated at any of 30 desks, second student can be seated at any of 29 remaining desks etc. Order does matter because it is not the same weather John or Jane sits at the first desk. ` prod_(i=9)^30 i=9*10*11*cdots*29*30=6578691959627754430464000000`
b. If a phone number has 7 digits (without area code) then there are `10^7=10000000` possible phone numbers in a given area. Each of the 7 digits can be 0,1,...,8,9.
c. `10^7` per area code times 301 area codes equals 3010000000.
This is the same as asking how many permutations of those 9 letters are there. Obviously order of letters in a word matters. `9! =362880`
4) Order of questions doesn't matter only if a question is chosen or not. `((20),(8))=125970`
5) `((14),(3))=364` Lower number is exponent of y.
She can choose 2 vanilla chocolate in `((4),(2))=6` ways and she can choose 2 chocolate from 12 in `((12),(2))=66` ways. Hence probability is `6/66=1/11.` You could use permutations here as well but you would have to calculate with bigger numbers, of course result would be the same.
9) To simplify the task let's say the dices are rolled one by one (this actually doesn't matter but it's useful for better understanding of the problem). First dice is rolled and we get some number so we must get the same number an all four of the remaining dices and probability of getting a particular number when rolling a dice is `1/6` . Hence we have