Jenn randomly chooses a number J from `1,2,3, ,19,20`. Bela then randomly chooses a number B from `1,2,3,...... 19,20` distinct from J. The value of B-J is at least 2 with a probability that can be expressed in the form `m/n` where m and n are relatively prime positive integers. Find m + n.

The real value of x satisfying ""^(3)sqrt(20x +^(3)sqrt(20x+13))=13 can be expressed as a/b where a and b are relatively prime positive integers. Find the value of b ?

Let lim_(x rarr oo)x ln(e(1+(1)/(x))^(1-x)) equal (m)/(n) where m and n are relatively prime positive integ Find (m+n).

The value of sum_(k=0)^(oo)(k^(2))/(4^(k)) can be written as (m)/(n) where m and n are relatively prime positive integers.Then the value of (m+n) is

Mr. B has two fair 6-sided dice, one whose faces are numbered 1 to 6 and the second whose faces are numbered 3 to 8. Twice, he randomly picks one of dice (each dice equally likely) and rolls it. Given the sum of the resulting two rolls is 9, The probability he rolled same dice twice is (m)/(n) where m and n are relatively prime positive integers. Then the value of (m+n) is

If the area enclosed by the curve y^(2)=4x and 16y^(2)=5(x-1)^(3) can be expressed in the form (L sqrt(M))/(N) where L and N are relatively prime and M is a prime, find the value of (L+M+N) .

Let a,b,c,d are distinct numbers to be chosen from the set {1,2,3,4,5}. If the least possible positive solution for x to the system of equations ax+by=1cx+dy=2} can be expressed in the form (p)/(q) where p and q are relatively prime,then find the value of (p+q).