The value r for which `((30),(r))((15),(0))+((30),(r-1)) ((15),(1))+........+((30),(0))((15),(r))` is maximum is/are

the value of r for which .^(30)C_r.^(20)C_0+^(30)C_(r-1).^(20)C_1.....+^(30)C_0.^(20)C_r is maximum (A) 10 (B) 15 (C) 20 (D) 25

If ((n),(r )) "denotes " ""^nC_r then (a) Evalutae : 2^(15)((30),(0))((30),(1))-2^(14)((30),(1))((29),(14))+2^(13)((30),(2))((28),(13)).......-((30),(15))((15),(0)) ( b) Prove that : Sigma_(r=1)^(n) ((n-1),(n-r))((n),(r))=((2n-1),(n-1)) ( c) Prove that : ((n),(r))((r),(k))=((n),(k))((n-r),(r-k))

If ((n),(r)) denotes nC_r , then Evaluate 2^15((30),(0))((30),(15))-2^14((30),(1))((29),(14))+2^13 ((30),(2))((28),(13))...............-((30),(15))((15),(0))

The value of (30,0)(30,10)-(30,1)(30,11)+(30,2)(30,11)-.........+(30,20)(30,30), where (n,r)=nC_(r) is a.(30,10) b.(30,15) c.(60,30)d.(31,10)

The value of (30,0)(30,10)-(30,1)(30,11)+(30,2)(30,12)-........+(30,20) (30,30) , where (n,r)=nC_r is a. (30,10) b. (30,15) c. (60,30) d. (31,10)

The value of (30,0)(30,10)-(30,1)(30,11)+(30,2)(30,11)-.............+(30,20)(30,30) , where (n,r)=nC_r is a. (30,10) b. (30,15) c. (60,30) d. (31,10)

The value of (30,0)(30,10)-(30,1)(30,11)+(30,2)(30,12)-.............+(30,20)(30,30) , where (n,r)=nC_r is a. (30,10) b. (30,15) c. (60,30) d. (31,10)

sum_(r=1)^(15)r(.^(15)C_(r))/(.^(15)C_(r-1))

The value of sum_(r=0)^(40)rC(40,r)C(30,r)

The value of sum_(r=0)^(15)r^(2)((""^(15)C_(r))/(""^(15)C_(r-1))) is equal to