Question

If \[{}^8{C_r} - {}^7{C_3} = {}^7{C_2}\], then the value of \[r\] is equal to
A. 3
B. 4
C. 8
D. 6

Answer 1

Hint: In order to determine the value of \[r\] from the equation use the important rule of combination that \[{}^n{C_{\left( {r - 1} \right)}} + {}^n{C_r} = {}^{n + 1}{C_r}\] by considering n as 7 and r as 3 and later compare both the combination written on LHS and RHS of the equation and on further simplification we get the required value of \[r\].

Complete step by step solution:
 We are given an expression
 $ \Rightarrow {}^8{C_r} - {}^7{C_3} = {}^7{C_2}$
In order to solve this equation, transfer the term not containing $r$from LHS to RHS,
$ \Rightarrow {}^8{C_r} = {}^7{C_2} + {}^7{C_3}$
Applying the most useful rule of combination \[{}^n{C_{\left( {r - 1} \right)}} + {}^n{C_r} = {}^{n + 1}{C_r}\] by considering \[n\] as 7 and \[r\] as 3, now our equation becomes
$ \Rightarrow {}^8{C_r} = {}^8{C_3}$
Comparing both the combinations, we get
$r = 3$
Therefore, the value of $r = 3$, So, option (A) is correct.

Note: 1. Factorials of proper fractions or negative integers are not defined. Factorial n defined only for whole numbers.
2.Meaning of Zero factorial is senseless to define it as the product of integers from 1 to zero. So, we define it as $ 0! = 1 $ .
3.Don’t forget to cross-check your answer at least once as it may contain calculation errors.