Please answer all three questions and show your work Thank y

Please answer all three questions and show your work. Thank you.

For a six year period. Andre deposited $800 each quarter into an account paying 4.2% annual interest compounded quarterly. (Round your answers to the nearest cent.) How much money was in the account at the end of 6 wars? Show work. How much interest was earned during the 6 year period? Show work. Andre then made no more deposits or withdrawals, and the money in the account continued to earn 4.2% annual interest compounded quarterly, for 3 more years. How much money was in the account after the 3 year period? Show work. How much interest was earned during the 3 year period? Show work. A contest has 15 finalists. One finalist is awarded first prize, another finalist is awarded second prize, and another is awarded third prize. How many different ways could the three prizes be awarded? Show work. There is a collection of 13 books. 8 of the books are fiction and 5 of the books are non-fiction. As an assignment, a student must read 6 of the books over the summer. In how many ways can 6 of the 13 books be chosen? Show work. In how many ways can the 6 books be chosen, if 4 of the books must be fiction and 2 of the books must be non-fiction? Show work. If 6 books are selected at random from the collection of 13 books, what is the probability that 4 are fiction and 2 are non-fiction? Give answer as a fraction or as a decimal rounded to nearest ten-thousandth (4 places after decimal) Show work.

Solution

18. there are 13 books out of which 8 are fiction and 5 are non fiction.

the student must read 6 books

a) from 13 books 6 books can be chosen in ¹³C₆ ways=13!/(6!*(13-6)!)=1716 [answer]

b) out of those 6 books 4 must be fiction and 2 must be non fiction.

so he has to choose 4 books from 8 books and 2 books from 5 books

so total number of ways=⁸C₄*⁵C₂=700 [answer]

c) let A be the event that out of the 6 books chosen 4 are fiction and 2 are non fiction

so P[A]=[number of cases favourable to A]/[total number of cases]

           =[number of ways out of 6 books 4 are fiction and 2 are non fiction]/[number of ways to choose 6 books from 13 books]=⁸C₄*⁵C₂/¹³C₆=700/1716   [answer]

17. to award the three prizes to 3 persons out of 15 persons at first in ¹⁵C₃ ways. and then of the three persons selected who would get the first prize,who would get the second prize and who get the third prize can be selected in 3! ways.

so total number of ways=¹⁵C₃*3!=2730 [answer]

16. andre deposited $800 each quarter for 6 years.

now 6 years=6*4=24 quarters

interest is 4.2% annual compounded quarterly.

so quarterly compound interest is 4.2/4=1.05%

now the $800 that he deposited in very first quarter is compounded in all 24 quarters

the $800 that he deposited in the second quarter is compounded in 23 quarters

and in this way ...........the $800 that he deposited in the 24th quarter is compounded in 1 quarter.

a) hence the amount of money in the account after 6 years is

=800(1+1.05/100)²⁴+800(1+1.05/100)²³+......+800(1+1.05/100)¹

=800[(1+1.05/100)²⁴+(1+1.05/100)²³+..........+(1+1.05/100)¹]

=800(1+1.05/100)[(1+1.05/100)²³+(1+1.05/100)²²+..........+1]

=800*1.0105[(1.0105²⁴-1)/(1.0105-1)] [ it is a sum of geometric progression]

=800*1.0105*27.13398=$21935.110       [answer]

b) so total amount in the account is $21935.110

and he deposited $800*24=$19200

hence the interest in 6 years=$21935.110-$19200=$2735.11   [answer]

c) now no deposite or no withdrawl. the amount $21935.11 continued to earn 4.2% annual interest compounded quarterly for 3 years.

so here principal is $21935.11

amount in the account after 3 years is A=21935.11(1+4.2/(4*100))^4*3=21935.11(1+1.05/100)¹²=$24864.265 [answer]

d) hence the interest=amount after 3 years-principal amount=$24864.265-$21935.11=$2929.155 [answer]