We all face problems with P&C. But not anymore.

STONE COLD RATTLESNAKE

Factorial: the continued product of first n natural numbers is called the “n factorial” and is denoted by n! or

i.e. n! = 1*2*3…*(n-1)*n

4! = 1*2*2*4

n! is defined for positive integers only.

0! is defined as 1.

n! = n*(n-1)!

Exponent of prime number p in factorial of n (n!)

The exponent of prime number of 3 in 100! is 48.
It means 100! is divisible by 3⁴⁸

How do you find it?
Let p be a prime number and n be a positive integer. Then find the last integer in the sequence 1,2,…,n which is divisible by p.
Express this integer as [n/p]p.
[n/p] denotes the greatest integer less than or equal to n/p

In case of 3 (p) and 100 (n); [n/p] is 33 and n/p is 33 and 1/3.

Let E_p(n) denote the exponent of the prime p in the positive integer n. Then,

E_p(n!) = E_p(1.2.3…(n-1).n)

This will be equal to E_p(p.2p.3p…[n/p]p)
= [n/p]+ E_p(1.2.3...[n/p])

This process continues and we get the answer

E_p(n!) = [n/p] + [n/p²]+…+[n/p^s]
Where s is the largest positive integer such that p^s≤n≤p^s+1

Hence applying the formula to find exponent of prime 3 in 100!

E₃(100!) = [100/3] + [100/3²] + [100/3³] + [100/3⁴]
= 33+11+3+1 = 48

Note: remember the meaning of notation [100/3] or [n/p]

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*n ways.

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.

Permutations Definition and Theorems

Each of the arrangement which can be made by taking some or all of a number of things is called a permutation.

Theorem 1

Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is given by

n(n-1)(n-2)…(n-(r-1))

Notation: Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is denoted by the symbol P(n,r) or ⁿC_r.

Permutations under certain conditions

Three theorems

Theorem 1
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.^n-1C_r-1

Theorem 2

The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is, ^n-1C_r-1

Theorem 3

The number of all permutations of n different objects taken r at a time, when two specified objects always occur together is 2!(r-1) ^n-2C_r-2

Then P(n,r) = ⁿC_r = n(n-1)(n-2)…(n-(r-1))

Permutations of Objects not all Distinct

Theorems and Formulas

Theorem
The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p+q = n, is

n!/p!q!

Formulas based on the above theorem

1. The number of mutually distinguishable permutations of n things, taken all at a time, of which p1 are alike of one kind, p2 alike of second,…, pr alike of of rth kind such that p1+p2+…pr = n, is

n!/p1!p2!…pr!

2. The number of mutually distinguishable permutations of n tings, of which p are alike of one kind, q alike of second and remaining all are distinct is
n!/p!q!

3. suppose there are r things to be arranged, allowing repetitions. Let further p1,p2,…,pr be the integers such that the first object occurs exactly p1 times, the second occurs exactly p2 times, etc. Then the total number of permutations of these r objects to the above condition is

(p1+p2+…+pr)!/p1!p2!…pr!

Combinations and Difference between Combinations and Permutations

Each of the different selections made by taking some or all of a number of objects, irrespective of their arrangement is called a combination.

Difference between Combinations and Permutations

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. For example AB and BA are same as combinations, but different as permutations.

Associate the word selection for combinations and arrangement for permuations.

Notation and Theorem for Combinations

Notation

The number of all combinations of n objects, taken r at a time is denoted by C(n,r) or ⁿC_r.

ⁿC_r is defined when n and r are non-negative numbers.

Theorem

The number of all combinations of n distinct objects, taken r at a time is given by

ⁿC_r = n!/(n-r)!r!

Results from the theorem

ⁿC_r = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)

ⁿC_n =1

ⁿC₀ = 1

ⁿC_r = ⁿP_r/r!

Properties of C(n,r)

Properties of ⁿC_r or C(n,r)

1. ⁿC_r = ⁿC_n-r

Note: If x=y = n

ⁿC_x = ⁿC_y

2. Let n and r be non-negative integers such that r≤n. Then

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

3. Let n and r be non-negative integers such that 1≤ r≤n. Then
ⁿC_r = (n/r) ^n-1C_r-1

4. If 1≤ r≤n, then

n.^n-1C_r-1 = (n-r+1)ⁿC_r-1

5. ⁿC_x = ⁿC_y implies x+y = n

6. If n is even, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_n/2.

7. If n is odd, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_(n+1)/2 or ⁿC_(n-1)/2.

Hope u like it.

Theorem 2

P(n,r) = ⁿC_r = n!/(n-r)!

Theorem 3

The number of all permutations of n distinct things taken all at a time is n!.

Theorem 4

0! = 1

Exponent of prime number p in factorial of n (n!)

Fundamental principle of multiplication

Permutations Definition and Theorems

Permutations under certain conditions

Permutations of Objects not all Distinct

Combinations and Difference between Combinations and Permutations

Notation and Theorem for Combinations

Properties of C(n,r)

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