Permutation Problems, 3

Category: Algebra, Statistics

"Published in Newark, California, USA"

A passenger train consists of 3 baggage cars, 5 day coaches, and 2 parlor cars. In how many ways can the train be arranged if the 3 baggage cars must come up front?

Solution:

The given word problem is about permutation problem because it involves the number of ways in arranging the objects or things. 

This permutation type is different and it is called a Distinguishable Permutation. 

If a set of n objects consists of k different kinds of objects with n₁, objects of the first kind, n₂ objects of the second kind, n₃ objects of the third kind, and so on, where n₁ + n₂ + ......... + n_k = n, then the number of distinguishable permutations of these objects is 

Now, in the given problem, if

n = 10 train cars in total
n₁ = 3 baggage cars
n₂ = 5 day coaches
n₃ = 2 parlor cars

then, the number of ways in arranging the 10 train cars will be equal to

If the 3 baggage cars must come up front, then the number of ways will be equal to

You have to multiply the previous ways by 3! because the three baggage cars themselves can be arranged in different ways at the front. Therefore, the final answer is

 

Triple Integration, 2

Category: Integral Calculus

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

Integrate the given equation first with respect to z. Consider x and y are constants

Next, integrate the equation above with respect to y. Consider x as a constant, we have

The integral of zero with respect to any variable is always equal to a constant. At the above equation, we cannot substitute the value of the limits because there's no variable present in the equation. 

Therefore,

Did you notice that if the limits for the second integration is from - x to x, the value of definite integral will be equal to zero? In this case, the function or curve is symmetrical to y-axis. We can rewrite the second integration as follows

or

Use the second equation above so that it is easier for us to integrate the equation, we have

Therefore,