Permutation and Combination GATE Questions

 Permutation and Combination GATE Questions

1Q. How many five digit numbers with distinct digits can be formed using the digit 1, 2, 3, 4, 5?

Solution: The number of distinct arrangements of 5 digits is calculated using the factorial of the number of digits:

$$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$==========================================================================

2Q. How many three digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition of digits?

Solution: 

1. Choose the first digit: There are 5 options (1, 2, 3, 4, or 5).
2. Choose the second digit: After selecting the first digit, there are 4 remaining options.
3. Choose the third digit: After selecting the first and second digits, there are 3 remaining options.

Thus, the total number of three-digit numbers can be calculated as follows:

$$\text{Total} = 5 \times 4 \times 3 = 60$$==========================================================================

3Q. How many three-digit odd numbers can be formed using the digits 1, 2, 3, 4, 5, when each digit occurs at most once in any of the numbers?

solution:

1. Choose the last digit: The last digit can be one of the odd digits: 1, 3, or 5. This gives us 3 options for the last digit.

2. Choose the first digit: After choosing the last digit, we have to select the first digit. The first digit can be any of the remaining digits (4 choices remaining).

3. Choose the second digit: After selecting the first and last digits, we have 3 remaining digits available for the second digit.

Total Combinations

Now, we can calculate the total number of three-digit odd numbers as follows:

• Choosing the last digit: 3 choices (1, 3, or 5).
• Choosing the first digit: 4 remaining choices.
• Choosing the second digit: 3 remaining choices.

Putting it all together, the total number of three-digit odd numbers can be calculated as:

$$\text{Total} = 3 \times 4 \times 3 = 36$$==========================================================================