express each of the following expressions in the form nPr or in the form nCr

a. 27x26x25x24/4x3x2x1
b. 30x29x28x27x26

To express each of the expressions in the form nPr or nCr, we need to understand what these notations mean.

1. nPr (Permutation):

The notation nPr represents the number of ways to arrange a set of objects when the order matters, without repetition. It is calculated using the formula:

nPr = n! / (n - r)!

where "n" represents the total number of objects, and "r" represents the number of objects taken at a time.
The exclamation mark (!) represents the factorial, which means multiplying the number by all positive integers less than it down to 1.

2. nCr (Combination):

The notation nCr represents the number of ways to select a subset of objects from a larger set when the order does not matter, without repetition. It is calculated using the formula:

nCr = n! / (r! * (n - r)!)

Now, let's apply these formulas to the given expressions:

a. 27x26x25x24/4x3x2x1

To express this expression in the form nPr or nCr, we need to determine if the order of the numbers matters or not. If the order matters, we use nPr; otherwise, we use nCr.

In this case, the order of the numbers does not matter because the division cancels out the effect of order. Therefore, we can express this expression in the form nCr.

Using the formula for nCr:

nCr = n! / (r! * (n - r)!),

we have:
n = 27 + 26 + 25 + 24
r = 4

so,

nCr = (27+26+25+24)! / (4! * ((27+26+25+24) - 4)!)

b. 30x29x28x27x26

Similar to the previous expression, we need to determine if the order of the numbers matters or not.

In this case, since we are given a multiplication of numbers with no division, the order matters. Therefore, we can express this expression in the form nPr.

Using the formula for nPr:

nPr = n! / (n - r)!

we have:
n = 30 + 29 + 28 + 27 + 26
r = 5

so,

nPr = (30 + 29 + 28 + 27 + 26)! / ((30 + 29 + 28 + 27 + 26) - 5)!

27!/(27-4)!4!= 27 C 4