chetan2u
In the next Cricket World Cup T-20, Team A has decided to score 200 runs only through 4s and 6s. In how many ways can the Team A score these 200 runs?
A. 13
B. 15
C. 16
D. 17
E. 18
Kudos for BEST and SHORTEST Method..
Not sure why this question has been categorized as "Combinations". All it is asking you is to find out # of pairs of x and y for which the equation \(4x + 6y = 200\) holds true.
Solution:
We know that x and y have to be non-negative integers.
However, x can not be 0 or 1 as for those values y does not have integer solution (200/6, 196/6 are not integers).
So the minimum value of x for which y has an integer solution is 2 (and corresponding y = 192/6 = 32)
Also note that LCM for (4, 6) is 12, hence every 2
6s can be replaced by 3
4s.
So we increase x by 3 and decrease y by 2 to get the next pair and so on.
1st pair -> (2, 32)
2nd pair -> (5, 30)
3rd pair -> (8, 28)
...
nth pair -> (50, 0)
x => {2, 5, 8, ..., 50}
y => {32, 30, 28, ..., 0}
All we have to do is to use formula for AP on either values of x or values of y to get the value of n.
=> 50 = 2 + 3(n-1) [Last term = 50, First Term = 2; d = 3]
=> n = 17