I offered an approach here:
https://www.beatthegmat.com/in-how-many-ways-t90061.html
Tokens
-
GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
-
ronnie1985
- Legendary Member
- Posts: 626
- Joined: Fri Dec 23, 2011 2:50 am
- Location: Ahmedabad
- Thanked: 31 times
- Followed by:10 members
This is a Manhattan Challenge question and teh method provided by the expert is right.
Follow your passion, Success as perceived by others shall follow you
-
Ganesh hatwar
- Senior | Next Rank: 100 Posts
- Posts: 97
- Joined: Sun Jun 24, 2012 11:23 pm
Rahul@gurome wrote:Thanks for the wonderul explanationgoyalsau wrote:There are 100 tokens numbered from 1 to 100. In how many ways can two tokens be drawn simultaneously so that their sum is more than 100?
- 4950
5050
2500
2550
5000
I got an answer 2525 ,, as i failed to taken into account ( 51,51) etc
So i did not know to choose between C or D
Is it better to guess lesser number than higher in such situations
Please help
Regards
Careful!
This question is full of tricks and traps!
One trick is there are two categories, n ≤ 50 and n > 50 (as goyalsau mentioned). And another one is the tokens are drawn simultaneously! Thus (2, 99) and (99, 2) are basically same.
Now, let's proceed to solve the problem.
As mentioned earlier, let's divide the numbers in two categories.
Category 1: n ≤ 50
For 1 : 1 set -> (1, 100)
For 2 : 2 set -> (2, 99), (2, 100)
For 3 : 3 set -> (3, 98), (3, 99), (3, 100)
� �
For 50 : 50 set -> (50, 51), ... , (50, 99), (50, 100)
Total number of sets = 1 + 2 + 3 + ... + 50
Category 1: n > 50
For 51 : 50 set -> (50, 51), (51, 52), ... ,(51, 100) No (51, 51)
For 52 : 51 set -> (49, 52), (50, 52), ... ,(52, 100) No (52, 52)
For 53 : 52 set -> (48, 53), (49, 53), ... ,(53, 100) No (53, 53)
� �
For 100 : 99 set -> (1, 100), (2, 100) ... , (99, 100) No (100, 100)
Total number of sets = 50 + 51 + 52 + ... + 99
Therefore total number of sets = (1 + 2 + 3 + ... + 50) + (50 + 51 + 52 + ... + 99)
= (1 + 2 + 3 + ... + 99) + 50
= (99)*(99 + 1)/2 + 50
= 99*50 + 50
= 50*100
= 5000
Now for the second trap, all the set are counted twice.
Thus effective number of set = 5000/2 = 2500
The correct answer is C.
-
mehaksal
- Master | Next Rank: 500 Posts
- Posts: 122
- Joined: Tue Jul 10, 2012 9:12 am
- Thanked: 2 times
- Followed by:8 members
I need to brush my PnC concepts, I was about to follow a very bad approach..I demarcated the two sets, but then I thought of simple doing :-
51C1 * 50C1
and hence 2550..how is this approach wrong?
51C1 * 50C1
and hence 2550..how is this approach wrong?
GMAT/MBA Expert
-
Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Here's another approach.goyalsau wrote:There are 100 tokens numbered from 1 to 100. In how many ways can two tokens be drawn simultaneously so that their sum is more than 100?
A) 4950
B) 5050
C) 2500
D) 2550
E) 5000
Of the 2 selected numbers, one of them will be the smaller number and one will be the larger number.
Since the smaller can range from 1 to 99 (the smaller number cannot be 100), there are 99 cases to consider.
Note: for each case, we'll count the number of ways that the sum can be greater than 100.
Case 1: 1 is the smaller number, which means the larger number must be 100 (1 way for the sum to be greater than 100)
Case 2: 2 is the smaller number, so the larger number must be 99 or 100 (2 ways)
Case 3: 3 is the smaller number, so the larger number must be 98, 99 or 100 (3 ways)
Case 4: 4 is the smaller number, so the larger number must be 97, 98, 99 or 100 (4 ways)
.
.
.
Case 50: 50 is the smaller number, so the larger number must be 51, 52,..., 99 or 100 (50 ways)
Case 51: 51 is the smaller number, so the larger number must be 52, 53,..., 99 or 100 (49 ways)
Case 52: 52 is the smaller number, so the larger number must be 53, 54,..., 99 or 100 (48 ways)
Case 53: 53 is the smaller number, so the larger number must be 54, 55..., 99 or 100 (47 ways)
.
.
.
Case 98: 98 is the smaller number, so the larger number must be 99 or 100 (2 ways)
Case 99: 99 is the smaller number, so the larger number must be 100 (1 way)
So, when we add up the all of these cases we get (1+2+3+...+50)+(49+48+47+...+3+2+1)
To find these two sums, we'll use the fact that the sum of the positive integers from 1 to k equals k(k+1)/2
So, when we add all of the cases together we get 50(50+1)/2 + 49(49+1)/2 = 2500
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
-
Ankur87
- Master | Next Rank: 500 Posts
- Posts: 110
- Joined: Sun Aug 19, 2012 4:39 am
- Thanked: 9 times
- GMAT Score:640
goyalsau wrote:There are 100 tokens numbered from 1 to 100. In how many ways can two tokens be drawn simultaneously so that their sum is more than 100?
4950
5050
2500
2550
5000
simplest method :
Taking 2 tokens :
100+1
100+2
....
100+99
= 99 >> tokens
99+2
99+3
99+4
.....
99.98
=97 >> tokens
.
.
.
.
.
.
.
.
52+49
=3 >> tokens
51+50
=1 >> tokens
TOTAL TOKENS :
1+3+5+7......99
= 50/2(2*1 + 49*2)
= 25(2 + 98)
= 2500
thebigkats wrote:thanks Rahul. Probably your way was faster. this is how got to this solution (although took me full 3 min
for no 1: 100 (total 1)
2: 100, 99 (total 2)
3: 100, 99, 98 (total 3)
...
50: 100, .. 52, 51 (total 50)
51: 100..52, 50 (but 51, 50 was already counted so leave it) TOTAL = 49
52: 100..53, 51, 50, 49 (but 49, 50 and 51 are already counted so leave them). TOTAL = 48
53: 100..54, 53, 52, 51, 50, 49, 48 (but 53..48 are already counted so leave them) TOTAL = 47
...
99: 100.., 98, 97..2 (but last ones except 100 are already counted so leave them) TOTAL = 1
100: all are already counted so TOTAL=0
So for first 50, total = n(n+1)/2 = 50(50+1)/2 = 25*51=1275
from 52..10, total = n(n+1)/2 = 49 (49+1)/2 = 25*49=1225
Total = 1275 + 1225 = 2500
better than rahulGMAT/MBA Expert
-
Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
This is an advanced-level question. 700+gughanbose wrote:What's the difficulty level of this question ?
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
-
rajeshsinghgmat
- Master | Next Rank: 500 Posts
- Posts: 171
- Joined: Tue Jan 08, 2013 7:24 am
- Thanked: 1 times
For SUM to be 100 the following combinations are possible :
(1,99),(2,98).....(49,51),(50,50)
which comes to 50 different ways.
But the order of drawing tokens is also important.
so it can also be in this way :
(99,1),(98,2).....(51,49),(50,50)
which also counts to 50 different ways.
Therefore, Total no. of Ways = 50 X 50 = 2500
(1,99),(2,98).....(49,51),(50,50)
which comes to 50 different ways.
But the order of drawing tokens is also important.
so it can also be in this way :
(99,1),(98,2).....(51,49),(50,50)
which also counts to 50 different ways.
Therefore, Total no. of Ways = 50 X 50 = 2500
-
ganeshrkamath
- Master | Next Rank: 500 Posts
- Posts: 283
- Joined: Sun Jun 23, 2013 11:56 pm
- Location: Bangalore, India
- Thanked: 97 times
- Followed by:26 members
- GMAT Score:750
1+100____________________________ 1goyalsau wrote:There are 100 tokens numbered from 1 to 100. In how many ways can two tokens be drawn simultaneously so that their sum is more than 100?
4950
5050
2500
2550
5000
2+100, 2+99______________________ 2
.
.
50+100,....,50+51________________ 50
51+100,....,51+52________________ 49
.
.
98+100,98+99_____________________ 2
99+100___________________________ 1
Total = 49(50) + 50 = 2500
Every job is a self-portrait of the person who did it. Autograph your work with excellence.
Kelley School of Business (Class of 2016)
GMAT Score: 750 V40 Q51 AWA 5 IR 8
https://www.beatthegmat.com/first-attemp ... tml#688494
Kelley School of Business (Class of 2016)
GMAT Score: 750 V40 Q51 AWA 5 IR 8
https://www.beatthegmat.com/first-attemp ... tml#688494
-
nikhilgmat31
- Legendary Member
- Posts: 518
- Joined: Tue May 12, 2015 8:25 pm
- Thanked: 10 times
Can't we do it like this.
first number is selected among 1...50 in 50-1+1 = 50 ways
second number is selected among 100...51 in 100 - 51 +1 = 50 ways
total number of ways = 50 * 50 = 2500
first number is selected among 1...50 in 50-1+1 = 50 ways
second number is selected among 100...51 in 100 - 51 +1 = 50 ways
total number of ways = 50 * 50 = 2500
