Question

`((30)/(0)) ((30)/(10)) - ((30)/(1)) ((30)/(11)) + ((30)/(2)) ((30)/(12)) + …. + ((30)/(2)) ((30)/(30))` is equal to
A. `.^(30)C_(11)`
B. `.^(60)C_(10)`
C. `.^(30)C_(10)`
D. `.^(65)C_(55)`

Answer 1

`A={:((30),(0)):}{:((30),(10)):}-{:((30),(1)):}{:((30),(11)):}+{:((30),(2)):}{:((30),(12)):}-...+{:((30),(20)):}{:((30),(30)):}`
`:.A=""^(30)C_(0).""^(30)C_(10)-""^(30)C_(1).""^(30)C_(11)+""^(30)C_(2).""^(30)C_(12)-...+""^(30)C_(20).""^(30)C_(30)`
`="Coefficient of "x^(20)" in "(1+x)^(30)(1-x)^(30)`
`="Coefficient of "x^(20)" in "(1-x^(2))^(30)`
`="Coefficient of "x_(20)" in "underset(r=0)overset(30)sum(-1)^(r^(30))C_(r)(x^(2))^(r)`
`=(-1)^(10)""^(30)C_(10)" "["for coefficient of "x^(20)," put "r=10]`
`=""^(30)C_(10)`