Sets and Relations
Complex Numbers 33
Sequences and Series
Locus and Straight Line
Measures of Dispersion
Bivariate Frequency Distribution and Chi Square Statistic
Permutations and Combinations
- Introduction of Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When All Objects Are Not Distinct
- Circular Permutations
- Properties of Permutations
- Properties of Combinations
- nCr , nCn =1, nC0 = 1, nCr = nCn–r, nCx = nCy, then x + y = n or x = y, n+1Cr = nCr-1 + nCr
- When all things are different
- When all things are not different.
- Mixed problems on permutation and combinations.
Theorem: `"^n P_r`= `"^n C_r` r!, 0 < r ≤ n.
Proof: Corresponding to each combination of `"^nC_r`, we have r ! permutations, because r objects in every combination can be rearranged in r ! ways.
Hence, the total number of permutations of n different things taken r at a time is `"^nCr` × r!. On the other hand, it is P n r . Thus
`"^n P_r` =`"^n C_r` * r!, 0 < r ≤ n.
1) From above n!/(n-r)!= `"^n C_r` * r!, i.e., `"^n C_r`= n!/[r!(n-r)!]
In particular, if r= n, `"^n C_n`= n!/(n!0!)= 1
2) We define `"^nC_0` = 1, i.e., the number of combinations of n different things taken nothing at all is considered to be 1. Counting combinations is merely counting the number of ways in which some or all objects at a time are selected. Selecting nothing at all is the same as leaving behind all the objects and we know that there is only one way of doing so. This way we define `"^nC_0` = 1.
3) As `(n!)/[0!(n-0)!]`= 1= `"^nC_0`, the formula `"^n C_r`= `(n!)/[r!(n-r)!]` is applicable for r=0 also. Hence
`"^n C_r`= `(n!)/[r!(n-r)!], 0 < r ≤ n`.
4) `"^n C_n-r`= `(n!)/ [(n-r)! (n-(n-r))!]= (n!)/[(n-r)!r!]= ``"^n C_r`,
i.e., selecting r objects out of n objects is same as rejecting (n – r) objects.
5) `"^nC_a` = `"^nC_b` ⇒ a = b or a = n – b, i.e., n = a + b
Theorem: `"^nC_r` + `"^nC_r-1`= `"^(n+1)C_r`
Proof: We have `"^nC_r` + `"^nC_r-1= (n!)/[r!(n-r)!] + (n!)/[(r-1)!(n-r+1)!]`
= `(n!)/ [r*(r-1)!(n-r)!] + (n!)/[(r-1)!(n-r+1)(n-r)!]`
= `(n!)/[(r-1)!(n-r)!] [(1/r) + 1/(n-r+1)]`
= `(n!)/[(r-1)!(n-r)!] * (n-r+1+r)/[r(n-r+1)]`
= `"^(n+1) C_r`