Equations In Permutation and Combinations with

Equations In Permutation and Combinations with Introduction of the Anekwe's Method of Swapping Factorials. Uchenna Okwudili Anekwe Department of Physics, University of Science and Technology, Aleiro, Kebbi State, Nigeria, ([email protected])

Abstract Permutations and combinations is concerned with determining the number of different ways of arranging and selecting objects out of a given number of objects, without actually listing them. Permutation involves arrangement of objects in a definite order while Combination deals with selection of some or all of a number of different objects. In this Manuscript we are concerned about solving equations of permutation and combination, on how to determine n number of items and how it can be selected or arranged in r number of ways and also to introduce Anekwe’ s method of swapping factorials in the case of determining r number of items. .

Keywords Solve, equation, substituting

Remarks

1. Introduction

Use permutations if a problem calls for the number of

The study of permutations and combinations is concerned

arrangements of objects and different orders are to be

with determining the number of different ways of arranging

counted.

and selecting objects out of a given number of objects,

Use combinations if a problem calls for the number of ways

without actually listing them. A permutation is an

of selecting objects and the order of selection is not to be

arrangement of objects in a definite order.

counted. [1]

Combinations On many occasions we are not interested in

In this Manuscript we are concerned about solving

arranging but only in selecting r objects from given n

equations of permutation and combination, on how to

objects. A combination is a selection of some or all of a

determine n number of items and how it can be selected or

number of different objects where the order of selection is

arranged in r number of ways and also to introduce The

immaterial.

Anekwe’ s method of swapping factorials in the case of

There are some basic counting techniques which will be

determining r number of items.

useful in determining the number of different ways of arranging or selecting objects. The two basic counting

2. Methodology

principles are Multiplication principle: Suppose an event E can occur in

This section involves solving equations in permutation and

m different ways and associated with each way of occurring

combination and also to introduce Anekwe’ s method of

of E, another event F can occur in n different ways, then the

swapping factorials in the case of determining r number of

total number of occurrence of the two events in the given

items through worked examples.

order is m × n. Addition principle: If an event E can occur in m ways and another event F can occur in n ways, and suppose that both

Example 2.1

can not occur together, then E or F can occur in m + n ways.

Solve the equation 1

np5 = 12 nc3

ncr =

n(n − 1)(n − 2).....( n − 1 + r ) r!

Solution

 n!     n − 5!  = 12  n!     n − 3!3! 

(1)

n ! n − 3!3! n − 3!3! * = = 12 n − 5! n! n − 5!

(2)

np5 = 12* nc3 n(n − 1)(n − 2)(n − 3)(n − 4) = 12*

3!* n(n − 1)( n − 2)( n − 3)( n − 4)

By Cross multiplying both sides of equation (2) we’ ve

12(n − 5)! = n − 3!3!

n(n − 1)(n − 2) 3!

= 12* n(n − 1)( n − 2) (3)

(n − 3)(n − 4) = 2

 we’ ve 12(n − 5)(n − 4)(n − 3)(n − 2) = (n − 3)(n − 2)(6)

n 2 − 7 n + 12 = 2 (4)

12(n − 5)(n − 4) = 6

n 2 − 7 n + 10 = 0

(5)

n 2 − 5n − 2n + 10 = 0

2  n 2 − 9n + 20  = 1

(6)

2n − 18n + 40 = 1

(7)

n(n − 5) − 2(n − 5) = 0

2

(n − 2)(n − 5) = 0

2n − 18n + 39 = 0 2

(8)

n = 2 or n = 5

Using the general formula we’ ve

n=

−(−18)  (−18) 2 − 4*2*39 3*5 = 18  2*2 4

(9)

Example 2.2

 n=5 Or

n=4

.

Solve the following

(10)

np3 =6 nc3

But on substitution n=4 does not satisfy the equation

 n=5

Solution Alternatively

np5 = 12 nc3

np3 =6 nc3

np5 = 12* nc3 From the rule of permutation

npr = n(n − 1)(n − 2).....(n − 1 + r ) 2

n! ( n − 3) ! = 6 n! 3!( n − 3)!

(1)

(2)

4c r = 6cr +3

4! 6! = r !( 4 − r )! (r + 3)!( 6 − (r + 3) )!

n ! n − 3!3! * =6 n − 3! n!

(3)

4! 6! = r !( 4 − r )! (r + 3)!( 3 − r )! (4)

4!(3 − r )!(r + 3)! = 6!(4 − r )!r !

n − 3!3! −6 = 0 n − 3!

Using Anekwe’ s method of Swapping Factorials we’ ve

1

n − 3!3! = 6 (5) n − 3! n − 3!3!− 6(n − 3)! =0 n − 3!

Multiply all through the equation above by 1!

(n − 3)!3!− 6(n − 3)! = 0

 On simplifying the equation above we’ ve

(n − 3)!3!− 6(n − 3)! = 0

2r 2 − 24r + 36 = 0

6(n − 3)! = 6(n − 3)!

4(3 − r )(r + 3) = 6(4 − r ) r

Using general formula to solve the above equation we’ ve (6)

−(−24)  242 − 4*2*36 24  17 r= = 2*2 4

Since both the left hand side and the right hand side are equal we will set one of them to be equal to zero in order to

 r=2 or r=10

get the corresponding values of n.

 From the equation above we’ ve

6(n − 3)! = 0

Applications Of The Principles Permutation And Combination

(7)

(n − 3)(n − 2) = 0 n 2 − 5n + 6 = 0

The Principles Of Permutation And Combination has a lot of applications [2-33] amongst which we've its Applications (8)

in:

On solving the above equation we’ ve

n=3

Or

Of

 The Telephone Numbering System.

n=2

 Computer architecture  Databases and data mining.

But on substitution n=2 does not satisfy the equation

 Patern Analysis

 n=3

 Operations research.  Homeland security e.t.c  Communication

Example 2.3 Given that

networks,

cryptography

network security

4c r = 6cr +3 find the values of r.

 Computational molecular biology  Languages

Solution 3

and

[12] Doerge R.W. and G. A. Churchill. 1996. Permutation tests for multiple loci affecting a quantitative character, Genetics, 142: 285-294.

 Molecular biology  Scientific discovery  Simulation

[13] Kapralski, A. 1993. New methods for generation of permutations, combinations, and other combinatorial objects in parallel, Journal of Parallel and Distributed Computing 17: 315-326.

3. Conclusion

[14] Kaufman, C., R. Perlman and M. Speciner. 2003. Network Security: Private Communication in a Public World, 2nd Ed., Upper Saddle River, NJ, Prentice Hall.

From the examples solved above we’ ve seen how to solve equations of permutation and combination, on how to

[15] Kotz. D. 1995. A data-parallel programming library for education (DAPPLE), SIGCSE Bulletin, 27(1): 76-81.

determine n number of items and how it can be selected or arranged in r number of ways and also

introduced

[16] Kumar Verma, A. and M. Trimbak Tamhankar. 1997. Reliability-based optimal task-allocation in distributed database management systems, IEEE Transactions on Reliability, 46(4): 452-459.

Anekwe’ s method of swapping factorials in the case of determining r number of items and through worked example we have justify the validity of Anekwe’ s method of

[17] Liu, C.L. 1968. Introduction to Combinatorial Mathematics, Computer Science Series, New York, McGraw-Hill, Chapter 1.

swapping factorials.

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