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I'm posting the next set of medium/hard PS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers. Good luck!

1. A password on Mr. Wallace's briefcase consists of 5 digits. What is the probability that the password contains exactly three digit 6?

A. 860/90,000 B. 810/100,000 C. 858/100,000 D. 860/100,000 E. 1530/100,000

3. For the past k days the average (arithmetic mean) cupcakes per day that Liv baked was 55. Today Bibi joined and together with Liv they baked 100 cupcakes, which raises the average to 60 cupcakes per day. What is the value of k? A. 6 B. 8 C. 9 D. 10 E. 12

5. There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble to remain in the jar? A. 460 B. 490 C. 493 D. 455 E. 445

6. A pool has two water pumps A and B and one drain C. Pump A alone can fill the whole pool in x hours, and pump B alone can fill the whole pool in y hours. The drain can empty the whole pool in z hours, where z>x. With pumps A and B both running and the drain C unstopped till the pool is filled, which of the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool? A. \(\frac{yz}{x+y+z}\)

7. Metropolis Corporation has 4 shareholders: Fritz, Luis, Alfred and Werner. Number of shares that Fritz owns is 2/3 rd of number of the shares of the other three shareholders, number of the shares that Luis owns is 3/7 th of number of the shares of the other three shareholders and number of the shares that Alfred owns is 4/11 th of number of the shares of the other three shareholders. If dividends of $3,600,000 were distributed among the 4 shareholders, how much of this amount did Werner receive? A. $60,000 B. $90,000 C. $100,000 D. $120,000 E. $180,000

8. A set A consists of 7 consecutive odd integers. If the sum of 5 largest integers of set A is -185 what is the sum of the 5 smallest integers of set A? A. -165 B. -175 C. -195 D. -205 E. -215

11. In an infinite sequence 1, 3, 9, 27, ... each term after the first is three times the previous term. What is the difference between the sum of 13th and 15th terms and the sum of 12th and 14th terms of the sequence? A. 10*3^11 B. 20*3^11 C. 10*3^12 D. 40*3^11 E. 20*3^12

12. x, y and z are positive integers such that when x is divided by y the remainder is 3 and when y is divided by z the remainder is 8. What is the smallest possible value of x+y+z? A. 12 B. 20 C. 24 D. 29 E. 33

13. If \(x=\frac{(8!)^{10}-(8!)^6}{(8!)^{5}-(8!)^3}\), what is the product of the tens and the units digits of \(\frac{x}{(8!)^3}-39\)? A. 0 B. 6 C. 7 D. 12 E. 14

1. A password on Mr. Wallace's briefcase consists of 5 digits. What is the probability that the password contains exactly three digit 6?

A. 860/90,000 B. 810/100,000 C. 858/100,000 D. 860/100,000 E. 1530/100,000

Total # of 5 digit codes is 10^5, notice that it's not 9*10^4, since in a code we can have zero as the first digit.

# of passwords with three digit 6 is \(9*9*C^3_5=810\): each out of two other digits (not 6) has 9 choices, thus we have 9*9 and \(C^3_5\) is ways to choose which 3 digits will be 6's out of 5 digits we have.

\(P=\frac{favorable}{total}=\frac{810}{10^5}\)

Answer: B.

Bunuel, I did a mistake while calculating 10^5 - I calculated 9*10^4. I have a doubt : when the question says "5 digit" code, doesn't it mean that numbers start from 10000? I have seen some GMATClub questions were three digit codes start from 100. Can you please help? Thanks

Since we are talking about a 5-digit password and not a 5-digit number, then it can start with zero. For example a password can be 00123 or 00001.

In regards to question 13, how do you know that (8!)^2 will have two 0's at the end?

You figure it out by knowing that there is a 5 as a factor in the final number.

example 8! = 8*7*6*5...*1

Now if u know how many fives are there you will have that number of zeros in the end because at least that many number of 2 will there for sure ...

Now IF u want to figure out how may fives are there in (which is equivalent to finding how many zeros in the end of the number ) then simply keep dividing the factorial by 5 and adding it to the result until 5^x exceeds the factorial, i know i have used very confusing language but a example will simplify things for sure

how many zeros at the end of 312! answer :- 312!/5 + 312!/5^2 + 312!/5^3 = 62 + 12 + 2 (now note i stopped at 5^3, because 5^4 =25*25=625 which is more that 312 ) hence :- 62 + 12 + 2 =76 number of 5s in the factorial and same number of zeros in the end because these many number of 2s will obviously be there in factorial and will form 10s .

IF you are still confused I would suggest you to download Bunuel's math's notes and even if you are not confused i would still suggest you to do the same ,

Bunuel could you please provide us/me with the answer keys so that I can cross check thanks once again for the questions and a amazing post .

I'll provide OA's with detailed solutions after some discussion in a couple of days. If you need OA's asap pm me and I'll send you them. _________________

[quote2]9. If x and y are negative numbers, what is the value of \(\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}\)? A. 1+y B. 1-y C. -1-y D. y-1 E. x-y

Correction/suggestion/advice/confirmation --- needed Answer to 9th question If x and y are negative numbers, what is the value of \(\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}\)?[/b] Sqrtx^2/x-sqrt(-y*|y|) = |x|/x-|y|= -1-y

-y = +y and |y| no matter what, comes out as positive hence Sqrt (-y*|y| ) comes out as possitive y and sqrtx^2 =|x|= positive x , of course from your tone i get that i'm going wrong some where but

8. A set A consists of 7 consecutive odd integers. If the sum of 5 largest integers of set A is -185 what is the sum of the 5 smallest integers of set A? A. -165 B. -175 C. -195 D. -205 E. -215

x ;x+2;x+2*2;x+2*3....x+2*6

since the set has consecutive odd integers, then mean =median

mean of 5 largest integers=-185/5= -37 median of 5 largest integers is the 3d of the largest or the 5th of the whole set A= x+2*4 x+8=-37 ;x=-45

median of the first 5 small integers =-45+2*2=-41

the sum of the first small odd integers=number of integers*mean (or median)=5*median=5*(-41)=-205 answ is

13. If x=\frac{(8!)^{10}-(8!)^6}{(8!)^{5}-(8!)^3}, what is the product of the tens and the units digits of \frac{x}{(8!)^3}-39? A. 0 B. 6 C. 7 D. 12 E. 14

9. If x and y are negative numbers, what is the value of \frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}? A. 1+y B. 1-y C. -1-y D. y-1 E. x-y -x/x-|y|=-1-(-y)=y-1 since y<0 ; x<0

12. x, y and z are positive integers such that when x is divided by y the remainder is 3 and when y is divided by z the remainder is 8. What is the smallest possible value of x+y+z? A. 12 B. 20 C. 24 D. 29 E. 33

x/y the remainder is 3 . then the smallest x is 3

y/z the remainder is 8. then the smallest y is 8 and z is 9

3+8+9=20

answ is B _________________

Happy are those who dream dreams and are ready to pay the price to make them come true

10. If x^2<81 and y^2<25, what is the largest prime number that can be equal to x-2y? A. 7 B. 11 C. 13 D. 17 E. 19

-9<x<9 -5<y<5

-9<x<9 -10<2y<10 x-2y<9-(-10) x-2y<19

the answer is 17. D again the mystical letter for today heh

11. In an infinite sequence 1, 3, 9, 27, ... each term after the first is three times the previous term. What is the difference between the sum of 13th and 15th terms and the sum of 12th and 14th terms of the sequence? A. 10*3^11 B. 20*3^11 C. 10*3^12 D. 40*3^11 E. 20*3^12

1, 3, 9, 27 ok please pay attention to the fact that the term of sequence has 3 one less than the number of the term,i.e. the 3d term is 9, but the number has two 3s (one less than the term). the 4th term is 3*3*3 ,which has three 3s.

ok, then we have the following- the q. asks us to find out - (13th term+15th term) -(12th term +14th term)

(3^12+3^14)-(3^11-3^13)= 3^11(3-1)+3^13(3-1)= 2*3^11(1+3^2)= 20*3^11 the answer is B

p.s. also note, that answer choices B and D are leaders of the day heh _________________

Happy are those who dream dreams and are ready to pay the price to make them come true

Bunuel, could u please tell us which question is still unanswered or answered wrong by all of us? I want to think on such questions more, before u post solutions.

and btw, do u read our solutions or just only answers? I wonder whether my way of thinking was ok. in some cases I tried to use another method not to repeat others. I just wonder whether it worked, or it was just coincidence

I do read all the solutions and award +1 Kudos if there is at least one correct explanation in the post.

As for your other question: I don't see a solution for #7, so you can try this one. _________________

[quote2]9. If x and y are negative numbers, what is the value of \(\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}\)? A. 1+y B. 1-y C. -1-y D. y-1 E. x-y

Correction/suggestion/advice/confirmation --- needed Answer to 9th question If x and y are negative numbers, what is the value of \(\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}\)?[/b] Sqrtx^2/x-sqrt(-y*|y|) = |x|/x-|y|= -1-y

First of all, I would suggest you to put x = -2 and y = -3 and see what you get.

\(\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}\)

\(\frac{\sqrt{(-2)^2}}{-2}-\sqrt{-(-3)*|-3|}\)

\(\frac{2}{-2}-\sqrt{9}\) (because square root will always be positive)

-1 - 3 = -1 + y (since y = -3)

Coming to your solution:

I see you have everything correct till the last step: |x|/x-|y|

|x|/x will be -1 which is fine. What about |y|? Do you remember how we define mods? We say: |x| = x if x >= 0 |x| = -x if x < 0

Why? If x >= 0 e.g. say x = 5, |x| = x = 5 Instead, if x < 0 e.g. say x = -6, is |x| = x? No! mods are never negative. So |x| = -x = -(-6) = 6 (Since x itself is negative, mod of x is negative of x which becomes positive.)

Therefore, if we know that y is negative, what is the value of |y|? |y| = -y

8. A set A consists of 7 consecutive odd integers. If the sum of 5 largest integers of set A is -185 what is the sum of the 5 smallest integers of set A? A. -165 B. -175 C. -195 D. -205 E. -215

10. If x^2<81 and y^2<25, what is the largest prime number that can be equal to x-2y? A. 7 B. 11 C. 13 D. 17 E. 19

Notice that we are not told that \(x\) and \(y\) are integers.

\(x^2<81\) means that \(-9<x<9\) and \(y^2<25\) means that \(-5<y<5\). Now, since the largest value of \(x\) is almost 9 and the largest value of \(-2y\) is almost 10 (for example if \(y=-4.9\)), then the largest value of \(x-2y\) is almost 9+10=19, so the actual value is less than 19, which means that the largest prime that can be equal to \(x-2y\) is 17. For example: \(x=8\) and \(y=-4.5\).

13. If \(x=\frac{(8!)^{10}-(8!)^6}{(8!)^{5}-(8!)^3}\), what is the product of the tens and the units digits of \(\frac{x}{(8!)^3}-39\)? A. 0 B. 6 C. 7 D. 12 E. 14

Now, since \(8!\) has 2 and 5 as its multiples, then it will have 0 as the units digit, so \((8!)^2\) will have two zeros in the end, which means that \((8!)^2-38\) will have 00-38=62 as the last digits: 6*2=12.

Answer: D.

Although there is hardly anything left after your's explanations but still I'll post my approach, which might help few people out here (esp. on the # trailing 0's as you people have kept in the Math book)--

first I simplified the whole x value as it looks so clumsy/heavy: given \(x = (8!)^6 [(8!)^4 - 1]/ (8!)^3[(8!)^2 - 1] = (8!)^3[(8!)^2 + 1][(8!)^2 - 1]/[(8!)^2 - 1] so, (x/ (8!)^3) - 39 = [(8!)^2 + 1] -39 = (8!)^2 -38\) Now as per the Q we need to know unit & tenth digit of the above number- as we know factorials of numbers > 4 have trailing zero's so here we can determine how many trailing 0's in 8!. Since \(8 < 5^2\) so that means we have only 1 trailing 0 in here (alternately you can have manual calc in this case as well because it's small number). So \((8!)^2\) has 2 trailing 0's. Now it does not matter what hundredth position is becasue we need to subtract only 38- this would result in 6 in tenth pos and 2 in unit pos. so product is = 12...

P.S.- just for refrence # trailing 0's in factorial N is determined by- \(N/5^1+N/5^2+....+N/5^k (where 5^k < N)\)for each division we take integer part (quotient) and then add them up. this should give # trailing 0's _________________

5. There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble to remain in the jar? A. 460 B. 490 C. 493 D. 455 E. 445

Total ways to select 8 marbles out of 7+5=12 is \(C^8_{12}\); Ways to select 8 marbles so that zero red marbles is left in the jar is \(C^7_7*C^1_5\); Ways to select 8 marbles so that zero blue marbles is left in the jar is \(C^5_5*C^3_7\);

Hence ways to select 8 marbles so that at least one red marble and at least one blue marble to remain the jar is \(C^8_{12}-(C^7_7*C^1_5+C^5_5*C^3_7)=495-(5+35)=455\).

Answer: D.

I tried feagure out this way first but thought to calculate little differently-

Since there are a toatl of 12 Marbels and after picking the required # marbels we are left with 4 marbels in the Jar- so our task is determine out of these 4 in how many ways we can pick R, B marbels so that there exists at least 1 marbel of each type- (R-1,B-3), (R-2,B-2),(R-3,B-1)

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