q37

[Saumyojit]

In an examination, the maximum marks for each of three papers are 50 each. Maximum marks for the 4th paper is 100. Find the number of ways in which the candidate can score 60% marks in the aggregate.

CAndidate needs to score 150 .
x1+x2+x3+x4 =150​

(0+ x^0+x^1+...+x^50) + ( 0+ x^0+x^1+...+x^50)_+(0+ x^0+x^1+...+x^50) + (0+ x^0+x^1+...+x^100 )

(x^0+x^1+...+x^50)^ 3 + (x^0+x^1+...+x^100 )

As it is in GP, sum = [ 1(1- x ^51) ] ^3 / (1-x) ^3
sum = [1(1-x ^101) ] /(1-x)

:: [ 1(1- x ^51) ]^3 /(1-x) ^3 + [1(1-x ^101) ] /(1-x)


(1-x^51)^3 * (1-x)^-3 + (1-x^101) * (1-x)^-1



Now , we have to find coefficient of x^150


How to do? we have to use multinomial THEOREM which is (n+r-1) C r which gives coefficient of x^150 in the expansion of (1-x)^ (-n)

 

[pka]

In an examination, the maximum marks for each of three papers are 50 each. Maximum marks for the 4th paper is 100. Find the number of ways in which the candidate can score 60% marks in the aggregate.

CAndidate needs to score 150 .x1+x2+x3+x4 =150
Now , we have to find coefficient of x^150

Using the generating function [imath]{\left( {\sum\limits_{k = 0}^{50} {{x^k}} } \right)^3}\left( {\sum\limits_{k = 0}^{100} {{x^k}} } \right)[/imath], its expansion contains the term [imath]110551x^{150}[/imath] SEE HERE

 

[Saumyojit]

Using the generating function (∑k=050xk)3(∑k=0100xk){\left( {\sum\limits_{k = 0}^{50} {{x^k}} } \right)^3}\left( {\sum\limits_{k = 0}^{100} {{x^k}} } \right)(k=0∑50xk)3(k=0∑100xk), its expansion contains the term 110551x150110551x^{150}110551x150

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[Saumyojit]

(0+ x^0+x^1+...+x^50) + ( 0+ x^0+x^1+...+x^50)_+(0+ x^0+x^1+...+x^50) + (0+ x^0+x^1+...+x^100 )

a big mistake i have done by adding them it should be multiplication!

 

[Saumyojit]

=coefficient of x^150 in (1+x+...+x50)^3 (1+x+...+x100)
= coefficient of x^150 in (1−x^51)^3*(1−x^101)*​

(1 −x)^-4

=coefficient of x^150 in ​

(1−3x^51+3x^102−x^153)(1−x^101)(1−x)^−4
[leaving terms containing powers of x greater than 150]​

Then?

 

[pka]

The page in link does not load

Try now!

 

[Saumyojit]

its done