how many ways a batsman can score 200 if he takes only 2's 4's and 6's

If the no. of 6 is 0, the number of ways of scoring the runs is [200/4] + 1 = 51 ways -> for every value of 4, there's a corresponding value of 2.
Similarly, if the number of 6s is 1, the number of ways of scoring the 194 runs is [194/4] + 1 = 49 ways

We see that the pattern is as follows: 51, 49, 48, 46, 45, ....4, 3, 1
The numbers are of the form 3k or 3k+1

This can be divided into two patterns -> 1, 4, 7,....49 and 3, 6, ...51
There are 17 terms in the first series and 17 terms in the second series
The sum of the first series is 17/2 * (2*1 + 16*3) = 425
The sum of the second series is 17/2 * (2*3 + 16*3) = 459

Total number of ways = 884

Hi Jaggu Kumar,
Let the number of 2's be x, 4,s be y and 6's be z
2x+4y+6z = 100
Suppose z = 0
2x+4y = 200
x+2y = 100
(x,y) = (0,50), (2,49).........(100,0) = 51 solutions
Suppose z = 1
2x+4y = 194
x+2y = 97
(1,48)(3,47).....(97,0) = 49 solutions
In this way AP is generated
Total solutions = 51+49+47....+1 = 26*26 = 676