A letter is takenout at random from 'ASSISTANT and another is taken out from "STATISTICS'. The probability that they are the same letters is:

Options

A. a. `35/96`
B. b. `19/90`
C. c. `19/96`
D. d. none of these

Answer

To find the probability that a letter taken at random from the word "ASSISTANT" and another letter taken from "STATISTICS" are the same, we will follow these steps:

### Step 1: Identify the letters in both words
- The word "ASSISTANT" has the letters: A, S, S, I, S, T, A, N, T.
- The word "STATISTICS" has the letters: S, T, A, T, I, S, T, I, C, S.

### Step 2: Count the occurrences of each letter in both words
- For "ASSISTANT":
- A: 2
- S: 3
- I: 1
- T: 2
- N: 1

- For "STATISTICS":
- S: 3
- T: 3
- A: 1
- I: 2
- C: 1

### Step 3: Calculate the total number of letters in each word
- Total letters in "ASSISTANT" = 9
- Total letters in "STATISTICS" = 10

### Step 4: Calculate the probability for each common letter
We will calculate the probability for each letter that appears in both words.

1. **For letter A**:
- Probability from "ASSISTANT" = Number of A's / Total letters in "ASSISTANT" = 2/9
- Probability from "STATISTICS" = Number of A's / Total letters in "STATISTICS" = 1/10
- Combined probability = (2/9) * (1/10) = 2/90

2. **For letter S**:
- Probability from "ASSISTANT" = 3/9
- Probability from "STATISTICS" = 3/10
- Combined probability = (3/9) * (3/10) = 9/90

3. **For letter I**:
- Probability from "ASSISTANT" = 1/9
- Probability from "STATISTICS" = 2/10
- Combined probability = (1/9) * (2/10) = 2/90

4. **For letter T**:
- Probability from "ASSISTANT" = 2/9
- Probability from "STATISTICS" = 3/10
- Combined probability = (2/9) * (3/10) = 6/90

### Step 5: Sum the probabilities of all common letters
Now we will sum up the probabilities calculated for each letter:
- Total probability = (2/90) + (9/90) + (2/90) + (6/90) = (2 + 9 + 2 + 6) / 90 = 19/90

### Final Answer
The probability that the letters drawn from "ASSISTANT" and "STATISTICS" are the same is **19/90**.

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