Evaluate each of the following. a. 4 ! b. 7 ! c. 0 ! d. (6 !)/(2 !) e. (5 !)/(2 ! 3 !) f. (6 !)/

step 1 So in this problem, we're going to be working with factorials. What is a factorial? Whenever you

Evaluate each of the following. a. 4 ! b. 7 ! c. 0 ! d. (6...

Evaluate each of the following. a. $4 !$ b. $7 !$ c. $0 !$ d. $\frac{6 !}{2 !}$ e. $\frac{5 !}{2 ! 3 !}$$ f. $\frac{6 !}{4 !(6-4) !}$$ g. $(0.3)^{4}$$ h. $\left(\begin{array}{l}7 \\3 \end{array}\right)$ i. $(\begin{array}{l}5 \\ 2\end{array}\right)$ j . $(\begin{array}{l}3 \\ 0\end{array}\right)$ k. $\left(\begin{array}{l}4 \\ 1\end{array}\right)(0.2)^{1}(0.8)^{3}$ l. $\left(\begin{array}{l}5 \\ 0\end{array}\right)(0.3)^{0}(0.7)^{5}$

Evaluate each of the following.
a. $4 !$
b. $7 !$
c. $0 !$
d. $\frac{6 !}{2 !}$
e. $\frac{5 !}{2 ! 3 !}$$
f. $\frac{6 !}{4 !(6-4) !}$$
g. $(0.3)^{4}$$
h. $\left(\begin{array}{l}7 \\3 \end{array}\right)$
i. $(\begin{array}{l}5 \\ 2\end{array}\right)$
j . $(\begin{array}{l}3 \\ 0\end{array}\right)$
k. $\left(\begin{array}{l}4 \\ 1\end{array}\right)(0.2)^{1}(0.8)^{3}$
l. $\left(\begin{array}{l}5 \\ 0\end{array}\right)(0.3)^{0}(0.7)^{5}$

Key Concepts

Factorial

Factorial is a function defined for non-negative integers, where n! (read as ‘n factorial’) is the product of all positive integers less than or equal to n, with the special case that 0! is defined as 1. It is a fundamental concept in combinatorics and appears in formulas for permutations, combinations, and probability distributions.

Binomial Coefficient

The binomial coefficient, usually read as ‘n choose r’ and written as C(n, r) or (n r), is defined as n! divided by the product of r! and (n-r)! This coefficient represents the number of ways to choose r elements from a set of n distinct elements without regard to the order and plays a central role in the binomial theorem and probability theory.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent, where the base is raised to the power of the exponent. It is used to denote repeated multiplication of the base, and is crucial for expressing terms in sequences, series, and probability expressions, such as those found in binomial distributions.

Transcript

00:01 So in this problem, we're going to be working with factorials.

00:05 What is a factorial? whenever you see the exclamation mark here, it's going to mean you multiply the number by every number smaller than it until you get to one.

00:18 So 4 factorial is 4 times 3 times 2 times 1, which is 24.

00:28 7 factorial, same thing.

00:31 Going to be 7 times 6 times 5 times 4 times 3 times 2 times 1 which is equal to 5 ,040.

00:42 So factorials are simple.

00:45 It's just a lot of work.

00:49 Zero factorial.

00:51 So you might think that this is going to be zero.

00:54 However, zero factorial is defined as 1.

01:00 You're always going to have zero factorial.

01:04 Be equal to one.

01:06 The reason being that we use factorials in a probability context, which is why you're doing this right now.

01:14 And if i tell you there's 10 items in a bag, pick zero of them.

01:20 How many outcomes are you going to have? well, that's one outcome, the outcome where you don't do anything.

01:28 It might feel like it's no outcome, but it's really one outcome.

01:32 So that's why zero factorial is equal to 1.

01:39 Now, dividing factorials, this might seem like it's more work, but it's actually a little bit less work.

01:47 So let's write this one out.

01:49 So 6 times 5 times 4 times 3 times 2 times 1 over 2 times 1.

01:56 So we can cancel out a 2 and a 1.

01:59 And we're just left with the 6, 5, 4, and 3, which is equal to 5, 4.

02:04 360.

02:07 So with bigger numbers, this is going to save a lot of time, especially if you don't have a calculator.

02:15 So in this one, we have five times four times three times two times one over two times one times three times one.

02:25 So cancel out the free to one.

02:28 Now we just have five times four divided by 2, which is 10.

02:37 Now, for f, so let's start writing it out, 6 times 5 times 4 times 3 times 2 times 1, over 4 times 3 times 2 times 1.

02:50 So 6 minus 4 factorial, that's 2 factorial, so times 2 times 1.

02:56 And let's start canceling stuff out.

02:59 The 4, 3, 2, and 1 go away, and we're just left with 6 times 5.

03:04 Divided by 2, which is 15.

03:10 Now for g, this is just algebra 1 question that we're being given here.

03:18 This is getting you exposed to the notation for the binomial theorem, but you are just going to multiply point three times itself four times, which is 0 .9 times 0 .9, which is 0 .0081.

03:38 All right...

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