Counting and ratio problem

[Veronica]

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?
a. 5 to 4; b.3 to 2; c. 2 to 1; d. 5 to 1; e. 6 to 1

I solved this problem as follows:
C (10;5) = 14 * 18
C (10; 4) = 14 *15
--> the ratio of the number of 5-letter code words to the number of 4-letter code words = (14*18)/(14*15) = 6/5

Please explain why the answer is 6 to 1????


Company S produces two kinds of stereos: basic and deluxe. Of the stereos produced by Company S last month, 2/3 were basic and the rest were deluxe. If it takes 7/5 as many hours to produce a deluxe stereo as it does to produce a basic stereo, then the number of hours it took to produce the deluxe stereos last month was what fraction of the total number of hours it took to produce all the stereos?
a. 7/17; b.14/31; c.7/15; d.17/35; e.1/2

Please show me how to solve this problem quickly???

Thank you!!!

[limestone]

We must use permutation here as order does matter.
ABCDE is different from ACDEB, as "act" is to "cat"
So the no. of 5 letter-code 10P5 = 10*9*8*7*6
the no. of 4 letter-code 10P4 = 10*9*8*7
So the ratio is : 6 to 1.

[gmatmachoman]

for problem no 2:

Basic steroes : 2/3 X

Delux Steroes : 1/3 X

Time taken to produce one basic stereo : Y hrs

Time taken to produce one deluxestereo : 7/5hrs

Time taken to produce (2/3) X basic stereo : (2/3)X *Y hrs


Time taken to produce1/3 X stereo : 7/5 * 1/3 * X * Y hrs = 7/15 * X * Y

Ratio of Deluxe/ Total

7/15 XY/ ( 2/3 XY + 7/15 XY)

= 7/17

Pick A

[limestone]

Sorry, I missed your second question. I use plug-in method as below:
First, I see the fraction 2/3, so I decide to choose the total no. of stereos a number that is divisible to 3. This case I choose 3.
So the no. of basic ones: 2 -- delux one : 1
Time to make 2 basic ones: 2*1 (take 1 as plug-in time to make a basic stereo) = 2
Time to make 1 delux one: 1* 7/5 = 7/5
Total time taken: 2+7/5 = 17/5
The fraction needed is : 7/5 / 17/5 = 7/17
The answer is A.