the highest prime less than 50, that divides the binomial coefficient 100C50 ? a) 37, b) 31, c)47 d)43


2 years ago

Share

Answers : (13)

                                        

100! contains 47 and 94, 50!*50! contains 47 twice. Therefore, answer is 47

2 years ago
                                        

Hi Rituparna,


 


For the greatest prime number, say p.


3p < 100


 


Because 100C50 = 100!/(50!*50!)


Each of the 50! would cancel the prime numbers in 100! twice. So there has to be one more prime for it to divide 100C50.


So 3p<100. Only possibility p=31.


 


Option B.


 


Regards,


Ashwin (IIT Madras).

2 years ago
                                        

yeah, the anwer is 31.... ur logic is right .... actually there is one more method using Legendre theorem..... would u know it by any chance?

2 years ago
                                        

I am guessing 33 should be the required answer. Because exponent of 33 in 100! is 3 and in 50!^2 is 2. But that isnt in option.

2 years ago
                                        

yeah the answer is b), thanx, but there is one more way using Legendre .... would u know it by any chance?

2 years ago
                                        

I have never heard of Legendres theorem. But is it this one by any chance (i dont know the name)


 


Exponent of p in n! = [n/p] + [n/p2] + [n/p3] + ....


 


If yes, then i have used this.

2 years ago
                                        

Hi Rituparna,


 


Yes you can use Legendre Theorem here.


It states the highest power of a prime number "p" in N! would be = Σ[N/pk]. where k = 1,2,..... and [.] denotes greatest integer function.


For example the power of 5 in 100! would be [100/5] + [100/25] + [100/125] +.... = 20+4 = 24. ie 100! will have 5^24.


 


So in this case check the prime number, for which you get the exponent to be three or more.


Only 31 will satisy that condition.


 


Hence option (B) using Legendre Theorem !!.


 


Regards,


Ashwin (IIT Madras).

2 years ago
                                        

Hi Jit Mitra,


 


Yes that is one of Legendre Theorem.


But it is applicable only on prime numbers.


And 33 = 3x11 is not a prime number.


 


And hence 31 is the largest prime number.


 


Regards,


Ashwin (IIT Madras).

2 years ago
                                        

If one needs to get the exponent of 33, then one has to split it into prime numbers first and then get the exponent.


 


33=3x11.


 


exp of 3 would be [100/3] + [100/9] + [100/27] + [100/81] = 33+11+3+1 = 48.


And exp of 11 would be [100/11] = 9.


 


So exp of 33 would be the lesser of the two.


Hence exponent of 33 in 100! would be 9.


 


Regards,


Ashwin (IIT Madras).

2 years ago
                                        

ys the ans wuld b 47.


thats bcoz, 100C50 = (51*52*53*......*100)/50!   and hence any num having a factor in series in numerator wuld divide.


also as per given terms in ques, 47 wuld be the correct 1.

2 years ago
                                        

Thanks a lot!!

2 years ago
                                        

Thanks sir. I missed that.

2 years ago
                                        

actually i m a bit weak at these binomial cofficients nd all .


so i have a kind request to ashwin sir ..... sir can u pls give some 4-5 relevant nd good questions based on the same or a bit diffrent concept .. ??

2 years ago

Post Your Answer

More Questions On Algebra

Ask Experts

Have any Question? Ask Experts
Post Question
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!!
Click Here for details
a dice is rolled the probability of getting a number greater than 2 a odd number are they independent events if yes i cant see how practically
 
 
Ans: Hello Student, Please find answer to your question below Sample space, S is S = {1, 2, 3, 4, 5, 6} (i) Let E be the event of occuring a number on die greater than 2 E = {3, 4, 5, 6}...
  img
Jitender Singh one month ago
 
Two events are independent if the occurence of one event does not effect the probability of other. Lets say 1 st event to be A and 2 nd to be B. Let C to be intersection of A and B. A and B...
 
Jitender Singh one month ago
 
no, i asked whether they are independent and how can this happen that the two events in single dice is independent
 
rohan one month ago
find the sum to n trms of the A.G.P 3+5/4+7/4 2 +9/4 3 +...
 
 
Take the LCM of the denominator and keep it outside of the series. Now from the nth term of the series, simplify and get the 2 GP’s in addition. Solve for the sum of the GP to get the...
  img
Ravi one month ago
 
the sum of n terms of A.G.P. To derive it call the sum of series be S n . Then multiply it by “r” of the G.P. and then subtract S n with rS n
  img
Aziz Alam one month ago
total number of seven digits ,, sum of whose digits is even .. (repitition is allowed)
 
 
Hello student, Please find the answer to your question below There are 9000000 seven digit numbers exist. (because smallest 7 digit number is 1000000 greatest 7 digit number is 9999999 So...
  img
SHAIK AASIF AHAMED 2 months ago
find dy/dx when y= s in^-1(2^(x+1)/ 1+ (4)^x)
 
 
Ans: Hello Student, Please find answer to your question below Using the chain rule
  img
Jitender Singh 4 months ago
vectors in detail
 
 
Vectors This is a vector: A vector has magnitude (how long it is) and direction : The length of the line shows its magnitude and the arrowhead points in the direction. You can add two...
  img
ruchi yadav 9 months ago
what is the value of i to the power i
 
 
Dear Student, i i = e ( /2)*i*i ................................................................................................................................. Thanks.
  img
Shobhit Varshney 3 months ago
View all Questions »