33Grid not clicking but got something in b/w
LHS is sum of no of ways of selecting k items from r items*ways of selecting n-k items from remaining (2n-r from 2n total items)........
where k=1,2,3......
so physically also it is just no of ways of selecting n from 2n as above in LHS in each sum term we are selecting k from r and n-k from remaining...2n-r so effictively its just selecting n frm 2n
62u have to make A+B moves.. of which "A" are alike.. moving in the forward direction.. and "B" alike moving upwards
So there are A+B objects to be arrranged. of which threre are 2 kinds. First kind move towards X axis. the 2nd Kind move towards Y axis.
Of which the first kind is repeated A number of times and 2nd B number of times
hence (A+B)!/A!.B! = A+BCA
33Not getting that "to go from (0,0) to (a,b)"
i earlier also tried this last yr... but forgot that (most probably not successful then also)
33Waise gr8, 3 solns for this one.....
Gud start 4 the Question of the day..
33that "to go from (0,0) to (a,b)" we have to go only frm x=integer and y= integer naa?
62Prerequisite: to go from (0,0) to (a,b) there are (a+b)Ca ways (Proof is not very tough.. just think it out!)
to reach from (0,0) to (n,n) the number of ways is 2nCn
to reach there u will have to pass through exactly one point on the red line..
The red line is x+y=r
so the point on this line will be (r-k,k)
no of ways to reach (n,n) passing through (r-k,k) will be
=no of ways to reach (r-k,k) and then (n,n)
=no of ways to reach (r-k,k) and then further (n-r+k,n-k) steps
rCk × 2n-rCn-k
Sum over all the possible values of k will give RHS!
1that was a gr8 question and even better solution.
looking forward to another such ques.
62Time's running out on this one! i will give another enhanced hint!
Almost close to the solution!
62hmm... but i have a far better proof.. which i wanted to showcase :)
good work.. btw :)
33That was easy..
coefficient of xn in (1+x)2n-r(1+x)r
that is in(1+x)2n..... 2nCn
as in (1+x)2n-r(1+x)r there is no xn for k>n in 2n-rCn-r
