There are 20 cards. Ten of these cards have the letter `"I"` printed on them and the other 10 have the letter `"I"` printed on them. If three cards picked up at random and kept in the same order, the probability of making word IIT is `1//9,1//3` b. `1//16 ,1//4` c. `1//4,1//2` d. none of these

Options

A. `4//27`
B. `5//38`
C. `1//8`
D. `9//80`

Explanation

To solve the problem, we need to find the probability of drawing the cards in the order that spells "IIT" from a set of 20 cards, where 10 cards have the letter "I" and 10 cards have the letter "T".

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have 20 cards in total: 10 cards with "I" and 10 cards with "T".
- We want to find the probability of drawing the cards in the order "IIT".

2. **Calculating the Probability of Each Draw:**
- **First Draw (I):**
- The probability of drawing an "I" first is the number of "I" cards divided by the total number of cards:
\[ P(\text{First I}) = \frac{10}{20} = \frac{1}{2} \]

- **Second Draw (I):**
- After drawing one "I", there are now 9 "I" cards left and a total of 19 cards remaining:
\[ P(\text{Second I | First I}) = \frac{9}{19} \]

- **Third Draw (T):**
- Now, we need to draw a "T". There are still 10 "T" cards left, and a total of 18 cards remaining:
\[ P(\text{Third T | First I and Second I}) = \frac{10}{18} = \frac{5}{9} \]

3. **Calculating the Total Probability:**
- The total probability of drawing the cards in the order "IIT" is the product of the individual probabilities:
\[ P(\text{IIT}) = P(\text{First I}) \times P(\text{Second I | First I}) \times P(\text{Third T | First I and Second I} \]
\[ P(\text{IIT}) = \frac{1}{2} \times \frac{9}{19} \times \frac{5}{9} \]

4. **Simplifying the Expression:**
- Now, we simplify the expression:
\[ P(\text{IIT}) = \frac{1 \times 9 \times 5}{2 \times 19 \times 9} = \frac{45}{342} \]
- Simplifying \(\frac{45}{342}\):
\[ \frac{45 \div 9}{342 \div 9} = \frac{5}{38} \]

5. **Final Answer:**
- The probability of drawing the cards in the order "IIT" is \(\frac{5}{38}\).
- Since this does not match any of the provided options, the answer is **d. none of these**.