Question

The value of `(""^9"C"_0 + ""^9"C"_1) + (""^9"C"_1 + ""^9"C"_2) + ... + (""^9"C"_8 + ""^9"C"_9)` is ______ 

Options

• 2¹⁰ - 2

• 2¹⁰ - 1

• 2¹⁰ + 1

• 2¹⁰

Answer 1

The value of `(""^9"C"_0 + ""^9"C"_1) + (""^9"C"_1 + ""^9"C"_2) + ... + (""^9"C"_8 + ""^9"C"_9)` is 2¹⁰ - 2.

Explanation:

`(""^9"C"_0 + ""^9"C"_1) + (""^9"C"_1 + ""^9"C"_2) + ... + (""^9"C"_8 + ""^9"C"_9)`

= `""^10"C"_1 + ""^10"C"_2 + ... + ""^10"C"_9 .....[∵ ""^n"C"_r + ""^n"C"_{r-1} = ""^{n + 1}"C"_r]`

= `(""^10"C"_0 + ""^10"C"_1 + ""^10"C"_2 + ... + ""^10"C"_9  + ""^10"C"_10) - ""^10"C"_0 - ""^10"C"_10`

= 2¹⁰ - 1 - 1 ............`[∵ ""^n"C"_0 + ""^n"C"_1 + ""^n"C"_2 + ... + ""^n"C"_n = 0^0]`

= 2¹⁰ - 2