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))`

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))

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

Evaluate: ((32)/(65)xx(13)/(16))-((-15)/(42)div(30)/(63))+((28)/(34)xx(-68)/(14))

The value of (3)/(15) +(13)/(14)-(19)/(21) +(31)/(35) -(23)/(30) is :

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)

Show that ""^(30)C_(0) +""^(30)C_(1)+""^(30)C_(2) +…+ ""^(30)C_(14) =2^(29)-(1)/(2) (""^(30)C_(15))

Prove that (2^(30)+2^(29)+2^(28))/(2^(31)+2^(30)-2^(29))=(7)/(10)