Official Quant thread for CAT 2013

@Sumi99 said:
Consider all possible seven-digit binary numbers having four 1s and three 0s. What is the sum of all such numbers? ???
0 0 0 1111>1+2+4+8
With one one
001 1101
001 1110
001 01 11
001 10 11

Same for 010,100
3*(1+2+4+8)+4*16=109
109+109+64+109+64+128
327+256=583

When two 1
101,110,011 we have 0011,0101,0110,1001,1010,1100
9(1+2+3+4)+12(16+32+64)
90+12*112
90+1344
1434

When three 1
111 we have 0001,0010,0100,1000
1+2+4+8+4*(16+32+64)
15+4*112
463

15+583+1434+463

rajiv lend out rs 9 to anni on condition that the amount is payable in 10 months by 10 equal installments of re. 1 each payable at the start of every month.what is the rate of interest per annum if the first instalment has to be paid one month from the date the loan is availed?

please suggest.

@Sumi99 said:
Consider all possible seven-digit binary numbers having four 1s and three 0s. What is the sum of all such numbers? ???
1 111 000
total = 6!/3!*3! = 20
left most digit is fixed

sum total considering numbers in binary
=
10^6 * 20 + 10^5*10 + 10^4*10 + 10^3*10 + 10^2*10+10*10+10
= 21111110


sum if they are converted in decimal,
2^6 * 20 + 10*( 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1) = 1910
@IIM-A2013 said:
rajiv lend out rs 9 to anni on condition that the amount is payable in 10 months by 10 equal installments of re. 1 each payable at the start of every month.what is the rate of interest per annum if the first instalment has to be paid one month from the date the loan is availed?please suggest.


General
lets say initial value is x and each instalment is y,rate of interest be r

say after 1 year u pay y ==> its inital value at the time of taking loan =y/r

after 2 years u pay y ==> its value at the time of taking loan=y/r^2

similarly...for others

here let monthly interest be r then annual interest will be 12r ,which would be the required answer.

1/r+1/r^2+1/r^3+.......1/r^10=9

solved it using wolfram

http://www.wolframalpha.com/input/?i=r^10%2B9r-8%3D0

whats the ans,btw?

@sujamait said:
1 111 000total = 6!/3!*3! = 20left most digit is fixedsum total considering numbers in binary=10^6 * 20 + 10^5*10 + 10^4*10 + 10^3*10 + 10^2*10+10*10+10= 21111110sum if they are converted in decimal,2^6 * 20 + 10*( 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1) = 1910
sujamait bhai.. please elaborate the highlighted part... not getting it.
@IIM-A2013 said:
rajiv lend out rs 9 to anni on condition that the amount is payable in 10 months by 10 equal installments of re. 1 each payable at the start of every month.what is the rate of interest per annum if the first instalment has to be paid one month from the date the loan is availed?please suggest.
I think it will a simple interest question (else calculations will be complex)

9(1 + 10r/1200) = (1 + 9r/1200) + (1 + 8r/1200) + (1 + 7r/1200) + .... + (1 + r/1200) + 1
9 + 90r/1200 = 10 + 45r/1200
r = 1200/45 = 80/3 = 26.66 %

@Sumi99 said:
Consider all possible seven-digit binary numbers having four 1s and three 0s. What is the sum of all such numbers? ???
All the numbers will be of form 1abcdef, where three of a, b, c, d, e, f are 1 and other three are 0.

So, C(6, 3) = 20 such numbers

Now, for every number 'abcdef', there will be a number '111111 - abcdef', which satisfies the required condition. So, sum will be 111111*10 (only for last 6 digits)

So, sum of all numbers = (2^6)*20 + 63*10 = 1280 + 630 = 1910
A similar question:-

Find the sum of all ten - digit binary strings having five zeros and five ones.
Note:- Strings can have a leading zero
@chillfactor said:
A similar question:-Find the sum of all ten - digit binary strings having five zeros and five ones.Note:- Strings can have a leading zero
total = 10c5 = 252

(2^9+2^8+2^7+...+2^0)*126
=(2^10-1)*126
=128898

@chillfactor said:
A similar question:-Find the sum of all ten - digit binary strings having five zeros and five ones.Note:- Strings can have a leading zero
126*1023= 128898?
@IIM-A2013 said:
rajiv lend out rs 9 to anni on condition that the amount is payable in 10 months by 10 equal installments of re. 1 each payable at the start of every month.what is the rate of interest per annum if the first instalment has to be paid one month from the date the loan is availed?please suggest.
As case is for periodic installment
We have to consider periodic interest on each installment given to rajiv by anni
Anni gave rs 1 for 9 months, another rs 1 for 8 months, another for 7 ...for 1 month and at last only rs 1
Total money received by rajiv=1*r*9/(12*100) + 1*r*8/(12*100) +...1*r*1/(12*100) + 10 = 10+(9*10/2)r/(100*12)=10+45r/1200---2
Anni should give 9+interest on rs 9 for 10 months(9*r*10/1200)---2
Equating both.
9+90r/1200=10+45r/1200
45r/1200=1
r=1200/45 pa
@chillfactor said:
A similar question:-Find the sum of all ten - digit binary strings having five zeros and five ones.Note:- Strings can have a leading zero
1111100000
1 in tenth+1 in 9th+.....+1 at last
Coz' 0 can appear at the beginning, for all cases, possible ways= 9!/4!*5!
9!/(4!*5!)*2^9+9!/(4!*5!)*2^8+...+9!/(4!*5!)*2^0

126(2^9+...2^1+1)=128898
@chillfactor said:
A similar question:-Find the sum of all ten - digit binary strings having five zeros and five ones.Note:- Strings can have a leading zero
128898

The number 1 occurs at each place (9!/4!5!) times
Sum = 9!/(4!5!) * (1023) = 128898
10 boxes each has n balls, you have a spring balance which can measure accurate weights . Each ball is 100gm.Only one box contains lighter balls of 90gm each.You can take out any number of balls from any number of boxes.
m=number of minimum weighings
find m ,when n=
1)9
2)8
3)1
4)2
5)3
ďťż
@19rsb said:
10 boxes each has n balls, you have a spring balance which can measure accurate weights . Each ball is 100gm.Only one box contains lighter balls of 90gm each.You can take out any number of balls from any number of boxes.m=number of minimum weighings find m ,when n=1)92)83)14)25)3ďťż
n =?
@19rsb said:
10 boxes each has n balls, you have a spring balance which can measure accurate weights . Each ball is 100gm.Only one box contains lighter balls of 90gm each.You can take out any number of balls from any number of boxes.m=number of minimum weighings find m ,when n=1)92)83)14)25)3
n = 1.

take out p balls from from box no. p
and say kth box has 90 gm balls
So the final weight should be = 100*(1+2+3+4+5....10) - k*10 = 10*(55-k)

hence based on the final weight observed we can find the value of 'k'.

ATDH.
@19rsb said:
10 boxes each has n balls, you have a spring balance which can measure accurate weights . Each ball is 100gm.Only one box contains lighter balls of 90gm each.You can take out any number of balls from any number of boxes.m=number of minimum weighings find m ,when n=1)92)83)14)25)3
For n=9 only 1
For n=8 only 2
For rest(n=1) only 4
For n=2,3 only 3 Edit
???
@anytomdickandhary said:
n = 1.take out p balls from from box no. pand say kth box has 90 gm ballsSo the final weight should be = 100*(1+2+3+4+5....10) - k*10 = 10*(55-k) hence based on the final weight observed we can find the value of 'k'.ATDH.
sir can u plz explain a bit more.....here we have to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3).....
@grkkrg said:
n =?
to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3).....
@Torque024 said:
For 9 only 1For 8 only 2For rest only 3???
discuss ur approach plz......as I dont hav d OA for dis
@19rsb said:
sir can u plz explain a bit more.....here we have to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3)..... to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3)..... discuss ur approach plz......as I dont hav d OA for dis
3) 1

When n = 9
the number of balls in each box = 0,1,2,3,4,5,6,7,8,9
So 1 weighing of all the boxes together will give the answer.
Ideal sum = 100 * 36 = 3600
If box 1 has 90g balls, then weight = 3600
If box 2 has 90g balls, then weight = 3590
If box 3 has 90g balls, then weight = 3580
and so on..

This will be true for any n

@19rsb said:
sir can u plz explain a bit more.....here we have to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3)..... to find m........for different values of n(when,n= 9,n=8,n=1,n=2,n=3)..... discuss ur approach plz......as I dont hav d OA for dis
10 boxes are there each has n balls

if n=9

Make two set A(first 9 boxes),B(last box)

From set A take 1 ball from first box 2 balls from 2nd box ... 9 balls form 9th box

If first box contains 90grms balls then total weight(TW)=90+(2+3+4..9)*100

And hence from the value of TW we can find then defective box in set A, and hence for total
if set A doesn't contains defective then B that's 10th ball will be defective

So weighting required will be 1


If n=8

Make two sets A(first 8 boxes),B(last 2 boxes)

Apply same procedure for A to find defective box. Take out 1 balls from 1, 2 from 2nd and so on
If A don't have defective box, then B will have defective box(worst case)

Again a weighting have to be done to identify the defective among two boxes in B

So weighting required will be 2 edit

If n=1

Make 2 sets A(first 5) B(next5)
Find total weight of A if !=5*100 then B will have defective box else A
In any case problem will reduce to 5 boxes
1 weighting will reduce prblem to 5 boxes
2 weighting will reduce to either 2 or 3
Take worst 3
now 2 weightings will be required total of 4 weightings

If n=2 or If n=3 Edit

Make 2 sets A(first 5) B(next5)
Find total weight of A if !=5*100 then B will have defective box else A
In any case problem will reduce to 5 boxes
1 weighting will reduce prblem to 5 boxes
2 weighting will reduce to either 2 or 3
Take worst 3

from first take 1 ball, 2nd 2 balls and from 3rd three balls and measure the weight.
For n=2 take out 0 balls 1st, 1 from 2nd 3 from 3rd and so on.
Now here only 1 weighting will be required
So total 3 weightings

@grkkrg said:
3) 1When n = 9the number of balls in each box = 0,1,2,3,4,5,6,7,8,9So 1 weighing of all the boxes together will give the answer.Ideal sum = 100 * 36 = 3600If box 1 has 90g balls, then weight = 3600If box 2 has 90g balls, then weight = 3590If box 3 has 90g balls, then weight = 3580and so on..This will be true for any n
Follow your approach if n=1, that's each box will have only 1 ball