Permutations And Combinations

1. The number of arrangements of 'n' things taken 'r' at a time

${}^{n}{P_{r}}$ = P (n, r) = n (n-1) (n - 2) ....... (n- r +1) = $\frac{n!}{{(n - r)}^{!}}$ where r $\leq$ n
2. ${}^{n}{P_{n}}$ = n!
3. Number of permutations with repetitions of n distinct objects taken r at a time = $n^{r}$
4. Number of circular permutations of n different things taken all at a time = (n-1) !
5. Out of n things if p are exactly alike of one kind, q exactly alike of the second kind, r exactly alike of the third kind and the remaining things all different, then the number of permutations of n things taken all at a time = $\frac{n!}{p! q! r!}$

6. The number of selections of 'n' things taken 'r' at a time = n $C_{r}$ = $\frac{{}^{n}{P_{r}}}{r!}$ = $\frac{n!}{r! {(n - r)}^{!}}$ , r $\leq$ n

7. n $C_{0}$ = 1, n $C_{1}$ = n, n $C_{2}$ = $\frac{n (n - 1)}{2}$ , n $C_{3}$ = $\frac{n (n - 1) (n - 2)}{3}$ , .............., n $C_{n}$ = 1

8. n $C_{r}$ = n $C_{n - r}$ .

9. If n $C_{r}$ = n $C_{s}$ then either r= s or r+s = n
10. n $C_{r}$ + n $C_{r - 1}$ = (n+1) $C_{r}$
11. Number of combinations of n dissimilar things taken r at a time when p particular things always occur = (n-p) $C_{r - p}$

12. Number of combinations of n dissimilar things taken r at a time when p particular exclude = (n-p) $C_{r}$

13. Number of diagonals of a ploygon of n sides= n $C_{2}$ - n = $\frac{n (n - 3)}{2}$

14. Number of ways in which m + n things can be divided into two groups containing m and n things = $\frac{(m + n)!}{m! n!}$ , where m = n. No. of ways of in which the 2m things can be divided into 2 groups having equal number of objects is $\frac{2 m}{2 (m!)}$

15. When n is even, n $C_{r}$ is greatest when r= $\frac{n - 1}{2}$ or $\frac{n + 1}{2}$

16. $\frac{n C_{r}}{n C_{r - 1}}$ = $\frac{n - r + 1}{r}$ ; $\frac{n C_{r + 1}}{n C_{r}}$ = $\frac{n - r}{r + 1}$

17. If N= $P_{1}^{a}$ . $P_{2}^{b} . P_{3}^{c}$ ......... where $P_{1}$ , $P_{2}$ , $P_{3}$ ...... are distinct primes and a, b, c, ...... are +ve integers then, the number of divisors of N= (a+1) (b+1) (c+1) ...... (Including 1 and N).

The sum of the devisors is $\frac{P_{1}^{a + 1} - 1}{P_{1} - 1}$ . $\frac{P_{2}^{b + 1} - 1}{P_{2} - 1}$ . $\frac{P_{3}^{c + 1} - 1}{P_{3} - 1}$ .......