Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 5 Permutations and Combinations will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2A Permutations and Combinations Formulas

→ Fundamental principle: If a work W_{1}, can be performed in m different ways and another work W_{2} can be performed in n different ways, then the two works simultaneously can be performed in mn different ways.

→ If n is a positive integer, then n! = n {(n – 1)!} and 1! = 1.

→ We define 0! 1.

→ The number of permutations of n dissimilar things taken ‘r’ at a time is denoted by

^{n}P_{r} and ^{n}p_{r} = \(\frac{n !}{(n-r) !}\) for 0 ≤ r ≤ n

→ If n, r are positive integers and r ≤ n, then

^{n}P_{r}= n.^{(n-1)}P_{(r-1)}(if r > 1)^{n}P_{r}= n.(n – 1)^{(n-2)}P_{(r-2)}.(if r ≥ 2).

→ The number of permutations of n dissimilar things taken V at a time

- containing a particular thing is r.
^{(n-1)}P_{(r-1)} - not containing a particular thing is
^{(n-1)}P_{r} - containing a particular thing in a particular place is
^{(n-1)}P_{(r-1)}

→ If n, r are positive integers and r < n, then ^{n}P_{r} = ^{(n-1)}P_{r} + r.^{(n-1)}P_{(r-1)}

→ The sum of the r-digit numbers that can be formed using the given n’ distinct non-zero digits

(r ≤ n ≤ 9) is ^{(n-1)}P_{(r-1)} × (sum of all n digits) × (111…………….1)_{(rtimes)}

→ In the above, if ‘0’ is one among the given ‘n’ digits, then the sum is

^{(n-1)}P_{(r-1)} × (sum of the digits) × (111……………1)_{(rtimes)}

^{(n-1)}P_{(r-1)} × (sum of the digits) × (111………………1)_{(r-1)times)}

→ The number of permutations of n dissimilar things taken r’ at a time when repetitions are allowed (i.e., each thing can be used any number of times) is n^{r}.

→ The number of circular permutations of n dissimilar things is (n – 1) !.

→ In the case of hanging type circular permutations like garlands of flowers, chains of beads etc., the number of circular permutations of n things is \(\frac{1}{2}\) [(n – 1) !].

→ If in the given n things, p like things are of one kind, q alike things are of the second kind, r alike things are of the third kind and the rest are dissimilar, then the number of permutations (of these n things) is \(\frac{\mathrm{n} !}{(\mathrm{p} !)(\mathrm{q} !)(\mathrm{r} !)}\)

→ The number of combinations of n things taken ‘r’ at a time is denoted by ^{n}C_{r} and ^{n}C_{r} = \(\frac{n !}{(n-r) ! r !}\) for 0 ≤ r ≤ n

→ If n, r are integers and 0 ≤ r ≤ n, then ^{n}C_{r} = ^{n}C_{(n-r)}

→ ^{n}C_{0} = ^{n}C_{n}; ^{n}C_{1} = ^{n}C_{(n-1)}

→ The number of ways of dividing ‘m + n’ things (m ≠ n) into two groups containing m, n things is \(\frac{(m+n+p) !}{(m !)(n !)(p !)}\)

→ The number of ways if distributing mn things equally to m persons is \(\frac{(\mathrm{mn}) !}{(\mathrm{n} !)^{\mathrm{m}}}\)

→ If p alike things are of one kind, q alike things are of the second kind and r alike things are of the third kind, then the number of ways of selecting one or more things out of them is (p + 1) (q + 1) (r + 1) – 1.

→ If m is a positive integer and m = P_{1}^{α1}, P_{2}^{α2} …………. P_{k}^{αk} where p_{1}, p_{2}, ………….. , p_{k} are distinct primes and α_{1}, α_{2}, …………….. α_{k} are positive integers then the number of divisors of m is (α_{1} + 1) (α_{2} + 1) ………. (α_{k} + 1) (This includes 1 and m).