Permutations refer to the arrangement of objects in a specific order. When all objects are distinct, the number of linear arrangements of \( n \) objects is \( n! \) (n factorial). However, when some objects are identical, we divide by the factorials of the frequencies of the repeated items to avoid overcounting.
The formula for the number of linear arrangements of \( n \) items, where \( p \) items are alike of one kind, \( q \) items are alike of another kind, \( r \) items are alike of a third kind, and the rest are distinct, is:
Here, \( p + q + r + \text{rest} = n \).
Question 1: How many 12-letter words can be formed using the letters in the word “MISSISSIPPII”?
Solution:
Step 1: Identify the total number of letters and their frequencies.
Step 2: Apply the permutation formula for alike items:
Here, \( n = 12 \), \( p = 5 \) (for I), \( q = 4 \) (for S), \( r = 2 \) (for P), and \( 1! \) (for M) is 1.
Number of arrangements =
Step 3: Calculate the factorials.
Denominator = \( 120 · 24 · 2 = 5,760 \)
Number of arrangements = \( \frac{479,001,600}{5,760} = 83,160 \)
Answer: 83,160 words can be formed.
Question 2: How many 6-digit numbers can be formed using the digits 1, 3, 5, 7 if one digit is repeated exactly three times?
Solution:
We can solve this using two methods.
Method 1: Case-by-case analysis
Since one digit is repeated three times, the remaining three positions are filled by the other three digits (one each). The possible cases are:
For each case, we arrange 6 digits where one digit appears 3 times. The number of arrangements per case is:
Total arrangements = \( 4 · 120 = 480 \)
Method 2: Combinatorial approach
Step 1: Choose the digit to be repeated three times from {1, 3, 5, 7}.
Number of ways =
Step 2: Arrange the 6 digits (three of the chosen digit and one each of the other three).
Number of arrangements =
Total arrangements = \( 4 · 120 = 480 \)
Explanation of Method 2: We use to select which digit is repeated three times. Then, we calculate the arrangements of the 6 positions, accounting for the three identical digits using the permutation formula.
Answer: 480 numbers can be formed.
Question 3: How many 6-digit numbers can be formed using the digits 1, 3, 5, 7 if there are two pairs of digits?
Solution:
This means two digits are repeated twice each, and the remaining two digits appear once each. We solve using two methods.
Method 1: Case-by-case analysis
Choose two digits to form pairs (e.g., 1 and 3 repeated twice each). Possible pairs are:
For each pair, arrange 6 digits where two digits appear twice each. Number of arrangements per case:
Total arrangements = \( 6 · 180 = 1080 \)
Method 2: Combinatorial approach
Step 1: Choose 2 digits out of 4 to be repeated twice.
Number of ways =
Step 2: Arrange the 6 digits (two digits twice each, two digits once each).
Number of arrangements =
Total arrangements = \( 6 · 180 = 1080 \)
Answer: 1080 numbers can be formed.
Question 4: How many 7-letter words can be formed using the letters in “BANANA”?
Solution:
Step 1: Identify the total number of letters and their frequencies.
Step 2: Apply the formula:
Denominator = \( 6 · 2 · 1 · 1 = 12 \)
Number of arrangements = \( \frac{5,040}{12} = 420 \)
Answer: 420 words can be formed.
Question 5: How many 5-digit numbers can be formed using the digits 2, 4, 6 if one digit is repeated exactly four times?
Solution:
Method 1: Case-by-case analysis
One digit is repeated four times, and another digit appears once. Possible cases:
For each case, arrange 5 digits:
Total arrangements = \( 6 · 5 = 30 \)
Method 2: Combinatorial approach
Step 1: Choose the digit to be repeated 4 times:
Step 2: Choose the other digit:
Step 3: Arrange:
Total = \( 3 · 2 · 5 = 30 \)
Answer: 30 numbers can be formed.
Permutations of alike items are a fascinating topic in combinatorics. By understanding the formula and practicing with examples, you can master this concept. Whether you’re preparing for exams or exploring math for fun, these problems provide excellent practice.