Factorials
NOTATION :
Let n and r be two positive integers such that \(r\le n\) then he value of \(\frac{{n!}}{{\left( {n - r} \right)!}}\) is denoted by \({}^n{p_r}\)
i.e.,\({}^n{p_r} = \frac{{n!}}{{\left( {n - r} \right)!}}\)
e.g (i)\({}^6{p_r} = \frac{{6!}}{{\left( {6 - 2} \right)!}}\)
\(= \frac{{6!}}{{4!}}\) \(= \frac{{6 \times 5 \times 4!}}{{4!}}\) =30
NOTE (i) : \({}^n{p_0} = \frac{{n!}}{{\left( {n - 0} \right)!}}\)
\( = \frac{{n!}}{{n!}}\) \(\therefore {}^n{p_0} = 1\)
i.e.,\({}^6{p_0} = 1,{}^8{p_0} = 1,.......\) so on
NOTE(ii) : \({}^n{p_n} = \frac{{n!}}{{\left( {n - n} \right)!}}\)
\(= \frac{{n!}}{{0!}}\)
\( = n!\) since \(0! = 1\)
i.e \({}^6{p_6} = 6!,{}^8{p_8} = 8!,{}^{10}{p_{10}} = 10!...........\). so on
Observe the following
i)\(^4{p_2} = \frac{{4!}}{{\left( {4 - 2} \right)!}} = \frac{{4!}}{{2!}} = \frac{{4 \times 3 \times 2!}}{{2!}}\)
\({ = ^{10}}{p_5}\)
\( \Rightarrow r = 5\)
* If n and r two positive integers such that \(r\le n\), then the expression \(\frac{{n!}}{{\left( {n - r} \right)!r!}}\) is denoted by \(^n{C_r}\)
i.e.,\(^n{C_r} = \frac{{n!}}{{\left( {n - r} \right)!r!}}\)
e.g \(^6{C_2} = \frac{{6!}}{{\left( {6 - 2} \right)!2!}}\)
\(= \frac{{6!}}{{4!2!}}\)
\(= \frac{{6 \times 5 \times 4!}}{{4!.2}}\) =15
NOTE : (i) \(^6{C_0} = \frac{{6!}}{{\left( {6 - 0} \right)!0!}}\)
\( = \frac{{6!}}{{6!0!}} = 1\)
Similarly \(^7{C_0} = 1{,^8}{C_0} = 1\).......so on.
NOTE:(ii) \(^7{C_7} = \frac{{7!}}{{\left( {7 - 7} \right)!7!}}\)
\(= \frac{{7!}}{{0!.7!}} = 1\)
Similarly \(^8{C_8} = 1{,^{10}}{C_{10}} = 1\) ......so on.
\({\therefore ^n}{C_n} = 1\)
Properties of \(^n{C_r}\)
(i)\(^n{C_r}{ = ^n}{C_{n - r}}\)
(ii)\({}^n{C_r} = {}^n{C_s} \Leftrightarrow r = s\) (or) \(n = r + s\)
(iii)\({}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}\)
(iv)\(r.{}^n{C_r} = n.{}^{n - 1}{C_{r - 1}}\)
(v)\(\frac{{{}^{n + 1}{C_{r + 1}}}}{{{}^n{C_r}}} = \frac{{n + 1}}{{r + 1}}\)
\(\frac{{{}^n{C_r}}}{{{}^n{C_{r - 1}}}} = \frac{{n - r + 1}}{r}\)
Short trick formula to find the value of \({}^n{C_r}\) .
e.g\(^6{C_2} = \frac{{6!}}{{\left( {6 - 2} \right)!.2!}}\)
\(= \frac{{6!}}{{4!.2!}}\)
\(= \frac{{6 \times 5 \times 4!}}{{4! \times 2}}\)
\(= \frac{{6 \times 5}}{{1 \times 2}}\)
Similarly,\({}^7{C_3} = \frac{{7 \times 6 \times 5}}{{1 \times 2 \times 3}}\)
\({}^{12}{C_4} = \frac{{12 \times 11 \times 10 \times 9}}{{1 \times 2 \times 3 \times 4}}\)
\({}^{14}{C_{10}} = {}^{14}{C_4} = \frac{{14 \times 13 \times 12 \times 11}}{{1 \times 2 \times 3 \times 4}}\)
Maximum Value Of \({}^n{C_r}\)
We can observe that in the list of \({}^6{C_0},{}^6{C_1},{}^6{C_2},{}^6{C_3},{}^6{C_4},{}^6{C_5},{}^6{C_6}\) the maximum value is \({}^6{C_3}\) .
Also, in the list of \({}^5{C_0},{}^5{C_1},{}^5{C_2},{}^5{C_3},{}^5{C_4},{}^5{C_5}\) , the maximum value is either \({}^5{C_2}\) or \({}^5{C_3}\)
In general, when n is even, the maximum value of \({}^n{C_r}\) is \({}^n{C_{\frac{n}{2}}}\) and when n is odd, the maximum value of \({}^n{C_r}\) is \({}^n{C_{\frac{{n - 1}}{2}}}\) or \({}^n{C_{\frac{{r + 1}}{2}}}\).