How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?

(i) The continued product of first n natural numbers is called the " n factorial" and is denoted by n or $n!$.

(ii) $n!=1\times 2\times 3\times 4\times ...\times (n-1)\times n$

(iii) Factorials of proper fractions and negative integers are not defined.

(iv) $0!=1$

(v) $\frac{\left(2n\right)!}{n!}=1\cdot 3\cdot 5....\left(2n-1\right)2^n$

2. Fundamental Principle of Multiplication: 

If there are two jobs such that one of them can be completed in $m$ ways, and when it has been completed in any one of these $m$ ways, second job can be completed in $n$ ways, then the two jobs in succession can be completed in $m\times n$ ways.

3. Fundamental Principle of Addition:

If there are two jobs such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in m+ n ways.

4. Permutations/Arrangements:

(i) If $n$ is a natural number and $r$ is a positive integer such that $0\le r\le n,$ then ${\mathrm{ }}^n{P}_r=\frac{n!}{(n-r)!}$.

(ii) Permutation of $n$ distinct objects taken $r$ at a time$={}^{n}P_r$.

(iii) Permutation of $n$ distinct objects taken all at a time$={}^{n}P_n=$$n!.$

(iv) Permutation of $n$ objects taken all at a time of which $p$ are alike$=\frac{n!}{p!}$.

(v) Permutations of $n$ objects of which $p$ are alike of one kind, $q$ are alike of second kind and remaining all are distinct$=$n!p! q!.

(vi) Permutation of $n$ objects such that $p$ objects are never together$=n!-\left(n-p+1\right)!$.

(vii) Permutation of $n$ objects such that $p$ objects are always together$=\left(n-p+1\right)!$.

(viii) Gap method:

Arrangement of boys and girls such that no two boys are together can be found by arranging all the girls first and then arranging the boys in the gaps.

(ix) Permutation of $n$ objects taken $r$ at a time when repetition is allowed$=n^r$.

(x) Permutation of $n$ objects taken all at a time when repetition is allowed$=n^n$.

5. Combinations/Selections:

(i) If $n$ is a natural number and $r$ is a positive integer such that $0\le r\le n,$ then ${\mathrm{ }}^n{C}_r=\frac{n!}{(n-r)!r!}$.

(ii) Selection of $r$ objects out of $n$ distinct objects$={}^{n}C_r$.

(iii) Selection of $r$ objects from $n$ distinct objects of which $p$ objects are always included$={}^{n-p}C_{r-p}$.

(iv) Selection of $r$ objects from $n$ distinct objects of which $p$ objects are never included$={}^{n-p}C_r$

(v) Selection of at least one object from $n$ distinct objects$=2^n-1$.

(vi) Selection of at least one object from $n$ objects of which $m$ objects are alike, $n$ objects are alike and rest $p$ are distinct$=\left(m+1\right)\left(n+1\right)2^p-1$.

6. Properties related to ${}^{n}C_r$:

(i) ${}^{n}C_r={}^{n}C_{n-r}$

(ii) If ${}^{n}C_x={}^{n}C_y$, then $x=y$ or $x+y=n$

(iii) ${}^{n}C_r+{}^{n}C_{r-1}={}^{n+1}C_r$

(iv) $\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}$

7. Relation between ${}^{n}C_r$ and ${}^{n}P_r$:

  ${}^{n}P_r={}^{n}C_r·r!$