Students in Dorm Question

[abhi.genx7]

1) Of all the students in a certain dormitory, are first-year students and the rest are second-year students. If of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

2)If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

3)If the sequence x1, x2, x3, ..., xn, ... is such that x1 = 3 and xn+1 = 2xn - 1 for n ≥ 1, then x20 - x19 =

4)If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?


Can someone help me out with these questions ?

[Rahul@gurome]

2)If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
It should be 3^k and not 3k, I believe.
p = 1*2*3.....29*30.
P = (1*2*4*5*7.......29)*(3*6*9*12*15*18*21*24*27*30) = (1*2...29)*(3^14)*(1*2*1*4*5*2*7*8*1*10)....

The maximum value of k is 14.

Something seems to be missing in second line of first question.

[saurabhmahajan]

Abhi boss...

Question 1 is incomplete.

Question 4:
5-letter code word: 10c5 = 252
4-letter code word: 10c4 = 210
So ratio = 252/210 = 6/5

[Rahul@gurome]

3)If the sequence x1, x2, x3, ..., xn, ... is such that x1 = 3 and xn+1 = 2xn - 1 for n ≥ 1, then x20 - x19 =
We can see that
x1=3 = 2^1 + 1
x2=2(3)-1=5 = 2^2 + 1
x3=2(5)-1=9 = 2^3 + 1
x4=2(9)-1=17 = 2^4 + 1.

Or x19 = 2^19 + 1.
x20 = 2^20 + 1.
Or x20 - x19 = 2^20 - 2^19 = 2^19(2-1) = 2^19.

[abhi.genx7]

Thanks a lot for the replies , Here is the question 1 again

Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5th of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

This is it

[abhi.genx7]

Oh sorry , I did not read the Q1 properly .
Got the ans

[saurabhmahajan]

is the OA for this dormitory question 1/5th ?

[abhi.genx7]

Yes the OA is 1/5