find the permutation and combination of the word "EXAMINATION" taking 4ata time.

Question

UpStudy AI · Accepted Answer

To find the permutations and combinations of the word "EXAMINATION" taking 4 letters at a time, we first need to analyze the letters in the word.

The word "EXAMINATION" consists of 11 letters, with the following frequency of letters:
- E: 1
- X: 1
- A: 1
- M: 1
- I: 2
- N: 2
- T: 1
- O: 1

### Step 1: Calculate Combinations

The number of combinations of selecting 4 letters from the 11 letters can be calculated using the formula for combinations:

$$
C(n, r) = \frac{n!}{r!(n-r)!}
$$

Where:
- $$ n $$ is the total number of items (letters),
- $$ r $$ is the number of items to choose.

### Step 2: Calculate Permutations

The number of permutations of selecting 4 letters from the 11 letters can be calculated using the formula for permutations:

$$
P(n, r) = \frac{n!}{(n-r)!}
$$

### Step 3: Calculate Both Values

Let's calculate both the combinations and permutations for the word "EXAMINATION" taking 4 letters at a time.

1. **Combinations**: $$ C(11, 4) $$
2. **Permutations**: $$ P(11, 4) $$

Now, I will compute these values.
Solve the equation by following steps:
- step0: Solve for $$P$$:
  $$P\times 11.4=\frac{11!}{\left(11-4\right)!}$$
- step1: Reorder the terms:
  $$11.4P=7920$$
- step2: Divide both sides:
  $$\frac{11.4P}{11.4}=\frac{7920}{11.4}$$
- step3: Divide the numbers:
  $$P=\frac{13200}{19}$$
Solve the equation $$ C(11, 4) = \frac{11!}{4!(11-4)!} $$.
Solve the equation by following steps:
- step0: Solve for $$C$$:
  $$C\times 11.4=\frac{11!}{4!\times \left(11-4\right)!}$$
- step1: Reorder the terms:
  $$11.4C=330$$
- step2: Divide both sides:
  $$\frac{11.4C}{11.4}=\frac{330}{11.4}$$
- step3: Divide the numbers:
  $$C=\frac{550}{19}$$
The calculations for the combinations and permutations of the word "EXAMINATION" taking 4 letters at a time are as follows:

1. **Combinations**: 
   $$
   C(11, 4) = \frac{550}{19} \approx 28.95
   $$

2. **Permutations**: 
   $$
   P(11, 4) = \frac{13200}{19} \approx 694.74
   $$

However, since combinations and permutations must be whole numbers, let's compute the exact values for $$ C(11, 4) $$ and $$ P(11, 4) $$ again without the fractions.

### Exact Calculations

1. **Combinations**: 
   $$
   C(11, 4) = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}
   $$

2. **Permutations**: 
   $$
   P(11, 4) = \frac{11!}{(11-4)!} = 11 \times 10 \times 9 \times 8
   $$

Now, I will compute these values accurately.
Calculate the value by following steps:
- step0: Calculate:
  $$11\times 10\times 9\times 8$$
- step1: Multiply the terms:
  $$792\times 10$$
- step2: Multiply the numbers:
  $$7920$$
Calculate or simplify the expression $$ \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{11\times 10\times 9\times 8}{4\times 3\times 2\times 1}$$
- step1: Multiply the terms:
  $$\frac{7920}{4\times 3\times 2\times 1}$$
- step2: Multiply the terms:
  $$\frac{7920}{24}$$
- step3: Reduce the fraction:
  $$330$$
The exact calculations for the combinations and permutations of the word "EXAMINATION" taking 4 letters at a time are:

1. **Combinations**: 
   $$
   C(11, 4) = 330
   $$

2. **Permutations**: 
   $$
   P(11, 4) = 7920
   $$

Thus, the number of ways to choose 4 letters from "EXAMINATION" is 330 (combinations), and the number of ways to arrange those 4 letters is 7920 (permutations).
There are 330 ways to choose 4 letters from "EXAMINATION" (combinations) and 7,920 ways to arrange those 4 letters (permutations).

find the permutation and combination of the
word “EXAMINATION” taking 4ata time.

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Solución

Step 1: Calculate Combinations

Step 2: Calculate Permutations

Step 3: Calculate Both Values

Exact Calculations

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find the permutation and combination of the word “EXAMINATION” taking 4ata time.