Permutation Problems

Category: Algebra, Statistics

"Published in Newark, California, USA"

If polygons are labeled by placing letters at their vertices, how many ways are there of labeling (a) a triangle, (b) a quadrilateral, (c) a hexagon with the first 10 letters of the alphabet?

Solution:

The given word problem above is about permutations. Permutation is an arrangement of a number of objects in a definite order. To "permute" a set of objects means to arrange them in a definite order. The number of permutations of n things taken r at a time is given by the formula

where n! (read as n factorial) is equal to n(n -1)(n - 2)......3∙2∙1. Take note that the values of n and r must be zero and positive numbers only. 0! is equal to 1. 

Now, let's go back to the given problem and solve for the permutations of the given polynomials. 

(a) For a triangle, the number of ways to label the vertices with the first 10  letters of the alphabet are

(b) For a quadrilateral, the number of ways to label the vertices with the first 10  letters of the alphabet are

 (c) For a hexagon, the number of ways to label the vertices with the first 10  letters of the alphabet are

Binomial Theorem, 4

Category: Algebra

"Published in Newark, California, USA"

Find the term which does not contain x in the expansion for

Solution:

Consider the formula in getting the value of rth term

In this problem, we need only x and y in order to solve for the value of r where x in not involve in the binomial expansion. If x is not involve in the binomial expansion, then the exponent of x is 0. Any number (except zero) raised to zero power is always equal to one.

Since we need the exponents of x in order to solve for the value of r, then we can omit y and 2 as follows

Take the logarithm on both sides of the equation to the base x, we have

Therefore, there's no x at the 3rd term.Consider again the formula in getting the value of rth term

Substitute the values of n, r, x, and y in order to get the value at the 3rd term, we have

Binomial Theorem, 3

Category: Algebra

"Published in Newark, California, USA"

Find the middle term for

Solution:

To get the position value of the middle term, which is also the rth term, consider this formula:

where n is the exponent of a binomial. If n is odd number, then the middle term is only one term and if n is even number, then the middle terms are two terms (consecutively). From the given equation above, if n = 11, then the value of r will be equal to

In this case, we need to get the 6th term of the binomial. To get the value of the rth term of (x + y)ⁿ, the formula can be written as

where r > 1. The given formula above must be memorized at all time especially when you are studying about Binomial Theorem. If you don't remember the formula, then you have to use the Pascal's Triangle in order to get the coefficients. Unfortunately, not all people can remember the coefficients of Pascal's Triangle especially if the exponent is high. 

Let's go back to the given problem, if n = 11 and r = 6, then the value of the rth term or 6th term will be equal to