Question

The value of
¹⁶C₉ + ¹⁶C₁₀- ¹⁶C₆ - ¹⁶C₇ is

• A. 0
• B. 1
• C. ¹⁷C₁₀
• D. ¹⁷C₃

Answer 1

The Correct Option is A

The value of $^{16}C_9 + ^{16}C_{10} - ^{16}C_6 - ^{16}C_7$ is

We use the property $^nC_r = ^nC_{n-r}$.

• $^{16}C_9 = ^{16}C_7$
• $^{16}C_{10} = ^{16}C_6$

Therefore, $^{16}C_9 + ^{16}C_{10} - ^{16}C_6 - ^{16}C_7 = ^{16}C_7 + ^{16}C_6 - ^{16}C_6 - ^{16}C_7 = 0$

 

Alternatively, using Pascal's identity $^nC_r + ^nC_{r-1} = ^{n+1}C_r$:

$^{16}C_9 + ^{16}C_{10} = ^{17}C_{10}$

$^{16}C_6 + ^{16}C_7 = ^{17}C_7$

So the expression becomes $^{17}C_{10} - ^{17}C_7 = ^{17}C_7 - ^{17}C_7 = 0$

Answer: (A) 0