PS - 2 questions

[cardoc]

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



If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n + 1)(n + 2) will be divisible by 8?

A. 1/4
B. 3/8
C. 1/2
D. 5/8
E. 3/4

[rijul007]

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

No of 5 letter words => x = 10P5 => 10!/5!
No of 4 letter words => y = 10P4 => 10!/6!

x/y = 6!/5! = 6

Option E

[rijul007]

If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n + 1)(n + 2) will be divisible by 8?

A. 1/4
B. 3/8
C. 1/2
D. 5/8
E. 3/4

If n is odd,

n(n+1)(n+2) will be divisible by 2 or 4


If n is even...
it will be divisible by 8

You can check by plugging in numbers...

No of even integers = 48
Total no of integer = 96
Probability = 1/2


Option C

[user123321]

if n is even then n(n+1)(n+2) is divisible by 8.using trial and error method you can get that.
so no of n's = 48 if n is even.

but we need to see if there is any odd number n which makes above condition true. yes what if n+1 is div by 8...then n is odd...so n can be 7,15,22,....95
no of odd n's possible = (95-7)/8 + 1 = 12

so we have 48+12 = 60 possibilities out of 96 => 10/16 = 5/8

[shankar.ashwin]

Try to identify a pattern here. Since 'n' is between 1 and 96, and you know you multiply 3 consecutive integers,

Start with (1)(2)(3) - Not div by 8
(2)(3)(4) - div by 8
(3)(4)(5) - not div by 8
(4)(5)(6) - div by 8
(5)(6)(7) - not div by 8
(6)(7)(8) - div by 8, and
(7)(8)(9) - div by 9.

SO you get all even numbers and also numbers one less than multiples of 8. 48 even numbers and 12 numbers one less than multiple of 8. So total 60.

60/96 = 5/8