What is the value of C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30) ?

What is the value of C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30) ? Correct Answer C(51, 51) - C(51, 0)

Concept: 

Formulae 

• nCr ​= nC_n-1​

Calculation:

Given,

C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30)

We know that nCr ​= nCn-r

So we can write,

C(51, 21) = C(51, 30)

C(51, 22) = C(51, 29)

C(51, 23) = C(51, 28)

C(51, 24) = C(51, 27)

C(51, 25) = C(51, 26)

So the above expression can be written as 

C(51, 30) - C(51, 29) + C(51, 28) - C(51, 27) + C(51, 26) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30)

= C(51, 30) - C(51, 30) + C(51, 29) - C(51, 29) + C(51, 28) - C(51, 28) + C(51, 27) - C(51, 27) + C(51, 26) - C(51, 26)

= 0

Checking option C:

C(51, 51) - C(51, 0)

Using nCr ​= nC_n-1 we can write C(51, 51) = C(51, 0),

C(51, 0) - C(51, 0) = 0

∴ The value of the expression is C(51, 51) - C(51, 0).