SOLUTION: Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE.. Please advise Q.2 1. THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETH

Algebra ->  Sequences-and-series -> SOLUTION: Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE.. Please advise Q.2 1. THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETH      Log On


   



Question 965989: Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE.. Please advise
Q.2 1. THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETHER OF WHICH A OBJECTS ARE SAME TYPE AND B OBJECTS ARE OF SAME TYPE AND REMAINING ARE OF SAME TYPE IS n! / a!b!(n-a-b)! ? TRUE / FALSE Please advise
Thanks well in advance.
Regards- Raheeq

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!


Q.1 IF THE RATIO OF AM AND GM OF TWO NUMBER IS 5:4. THE RATIO BETWEEN THE NUMBERS IS 4:1? TRUE / FALSE..
here's a definition of geometric mean:
http://www.mathwords.com/g/geometric_mean.htm

here's a definition of arithmetic mean:
http://www.mathwords.com/a/arithmetic_mean.htm

based on those definitions, you get the following;

AM of a and b is (a+b)/2

GM of a and b is sqrt(a*b)

the ratio of AM to GM is equal to AM/GM which is equal to 5/4.

this means that AM = 5 and Gm = 4, or any multiple of that.

for example, if AM = 10, then GM = 8.

the ratio is still 5/4.

we'll set AM = 5 and GM = 4 and solve using those values.

AM = (a+b)/2.

since AM = 5, this means that (a+b)/2 = 5.

solve for a to get a = 10 - b.

GM = sqrt(a*b).

since GM = 4, this means that sqrt(a*b) = 4

since a = 10 - b, then sqrt(a*b) = 4 becomes sqrt((10-b)*b) = 4.

square both sides of this equation to get (10-b)*b = 16

simplify to get 10b - b^2 = 16

add b^2 to both sides of the equation and subtract 10b from both sides of the equation to get 0 = b^2 - 10b + 16

factor the right side of the equation to get 0 = (b-8) * (b-2)

solve for b to get b = 8 or b = 2.

go back to the first equation of (a+b)/2 = 5.

multiply both sides of this eqution to get a+b = 10.

when b = 8, solve for a to get a = 2.

when b = 2, solve for a to get a = 8.

that should do it.

you either have a = 2 and b = 8, or you have a = 8 and b = 2.

let's first work with a = 2 and b = 8.

the ratio of (a+b)/2 divided by sqrt(a*b) is equal to 5/4.

the ratio of b/a is 8/2 which simplifies to 4/1.

when a = 8 and b = 2, then the ratio of (a+b)/2 divided by sqrt(a*b) is still equal to 5/4.

the ratio of a/b is 8/2 which simplifies to 4/1.

looks like your first statement is true.

the ratio of the larger nmber to the smaller number is 4 to 1 in both cases.

your first statement is true.

now to your second statement.

Q.2 1. THE NUMBER OF PERMUTATION OF N OBJECTS TAEKN TOGETHER OF WHICH A OBJECTS ARE SAME TYPE AND B OBJECTS ARE OF SAME TYPE AND REMAINING ARE OF SAME TYPE IS n! / a!b!(n-a-b)!

you have a object that are the same, and you have b objects that are the same and you have c objects that are the same, where c is equal to n - a - b.

the number of permutations will be n! / (a! * b! * c!).

this is the same as n! / (a! * b! * (n-a-b)!).

the statement appears to be true.

we can test it out using a simple example.

suppose you have 4 objects.

let's call the objects a,b,c,d.

since they're all different, the number of permutations of that will be 4! = 24.

the permutations are:

abcd
abdc
acbd
acdb
adbc
adcb

bacd
badc
bcad
bcda
bdac
bdca

cabd
cadb
cbad
cbda
cdab
cdba

dabc
dacb
dbac
dbca
dcab
dcba

now we'll translate b into a, and d into c, so we have a,a,c,c.

the number of permutations will be reduced because there will be duplicates that have to be eliminated.

the new set of the same 4 objects that contains the duplicates will be:

aacc
aacc
acac
acca
acac
acca

aacc
aacc
acac
acca
acac
acca

caac
caca
caac
caca
ccaa
ccaa

caac
caca
caac
caca
ccaa
ccaa

this set will now be sorted so that all duplicates will be shown next to each other.

aacc
aacc duplicate
aacc duplicate
aacc duplicate
acac
acac duplicate
acac duplicate
acac duplicate
acca
acca duplicate
acca duplicate
acca duplicate
caac
caac duplicate
caac duplicate
caac duplicate
caca
caca duplicate
caca duplicate
caca duplicate
ccaa
ccaa duplicate
ccaa duplicate
ccaa duplicate


all the duplicates will now be removed and all that will be left are the permutations that are unique.

aacc
acac
acca
caac
caca
ccaa

the set of 24 has been reduced to 6.

this is equal to 4! / (2! * 2!) which is equal to 24/4 which is equal to 6.

the same applies to your larger problem.

you've got n objects.

you've got a that are the same and b that are the same and c that are the same.

the number of permutations will be n! / (a! * b! * c!).

if c happens to be (n - a - b), then the formula becomes n! / (a! * b! * (n-a-b)!)

the statement is true.