Question

Find the value of \[\dfrac{{^n{C_{r - 1}}}}{{^n{C_r}}}\]

Answer 1

Hint: In the given question, we have to first find the coefficient for both numerator and the denominator and then write them in the form of the factorial. Now reduce the equation by cancelling out some same quantities. If the factorials are not in the same form try to bring them in the same form where \[r!\] can be written as \[r! = r \times \left( {r - 1} \right) \times \left( {r - 2} \right) \times \left( {r - 3} \right) \times ...........2 \times 1\] or can be written as \[r! = r \times \left( {r - 1} \right)!\]

Complete step by step answer:
We know \[{}^m{C_n} = \dfrac{{m!}}{{\left( {m - n} \right)!n!}}\], now we have to find the coefficient for \[\dfrac{{{}^n{C_{r - 1}}}}{{{}^n{C_r}}}\]
\[\dfrac{{{}^n{C_{r - 1}}}}{{{}^n{C_r}}} = \dfrac{{\dfrac{{n!}}{{\left( {n - r + 1} \right)!\left( {r - 1} \right)!}}}}{{\dfrac{{n!}}{{\left( {n - r} \right)!r!}}}}\]
Now simplify the expression
\[
  \dfrac{{{}^n{C_{r - 1}}}}{{{}^n{C_r}}} = \dfrac{{\dfrac{{n!}}{{\left( {n - r + 1} \right)!\left( {r - 1} \right)!}}}}{{\dfrac{{n!}}{{\left( {n - r} \right)!r!}}}} \\
   = \dfrac{{n!}}{{\left( {n - r + 1} \right)!\left( {r - 1} \right)!}} \times \dfrac{{\left( {n - r} \right)!r!}}{{n!}} \\
 \]
we know factorial of
\[r! = r \times \left( {r - 1} \right) \times \left( {r - 2} \right).......... \times \left( {r - n} \right)\]
We can also write as
\[
  r! = r \times \left( {r - 1} \right) \times \left( {r - 2} \right).......... \times \left( {r - n} \right) \\
   = r \times \left( {r - 1} \right)! \\
 \]
Now put r! in the expression:
  \[
  \dfrac{{{}^n{C_{r - 1}}}}{{{}^n{C_r}}} = \dfrac{{n!}}{{\left( {n - r + 1} \right)!\left( {r - 1} \right)!}} \times \dfrac{{\left( {n - r} \right)!r!}}{{n!}} \\
   = \dfrac{{n!}}{{\left( {n - r + 1} \right)!\left( {r - 1} \right)!}} \times \dfrac{{\left( {n - r} \right)! \times r \times \left( {r - 1} \right)!}}{{n!}} \\
   = \dfrac{{\left( {n - r} \right)! \times r}}{{\left( {n - r + 1} \right)!}} \\
 \]
Also \[\left( {n - r + 1} \right)!\] can be written as
\[\left( {n - r + 1} \right)! = \left( {n - r + 1} \right) \times \left( {n - r} \right)!\]
Hence,
\[
  \dfrac{{{}^n{C_{r - 1}}}}{{{}^n{C_r}}} = \dfrac{{\left( {n - r} \right)! \times r}}{{\left( {n - r + 1} \right)!}} \\
   = \dfrac{{\left( {n - r} \right)! \times r}}{{\left( {n - r + 1} \right) \times \left( {n - r} \right)!}} \\
   = \dfrac{r}{{\left( {n - r + 1} \right)}} \\
 \]

Additional Information: Binomial coefficient \[{}^m{C_n}\] is the number of ways of picking favorable m outcomes from the total number, n number of possibilities and is represented as \[{}^m{C_n} = \dfrac{{m!}}{{\left( {m - n} \right)!n!}}\] where factorial \[\left( ! \right)\] is the product of all-natural numbers less than the number including the number itself.
For example: \[5! = 5 \times 4 \times 3 \times 2 \times 1\].
\[{}^n{C_r}\], is the mathematical representation of the combination which is a method of selection of some items or all of the items from a set without taking the sequence of selection into consideration whereas in the case of permutation which is the method of arrangements of items of a set the sequence is considered represented as \[{}^n{P_r}\].

Note: In some special cases, when we have to find the \[0!\] factorial value it will be \[0! = 1\], and factorials are defined only for the non-negative numbers. It is a very crucial point to note here that, students confuse with the formulas of the combination and the permutation often. So, be wise while choosing the formulae of combination and permutation which is \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\] and \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\].