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Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is?
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Three ladies and three gents can be at around table so that any two an...
Understanding the Problem
To find the number of ways three ladies (L1, L2, L3) and three gents (G1, G2, G3) can sit around a table such that exactly two ladies sit together, we need to follow specific steps.
Step 1: Treating Two Ladies as One Unit
- Choose any two ladies to sit together. The combinations are:
- (L1, L2), (L1, L3), (L2, L3) → 3 ways.
- Treat the chosen pair as a single unit or block (LL). Now we have:
- 1 block (LL), 1 lady (L), and 3 gents (G1, G2, G3). Total = 1 + 1 + 3 = 5 units.
Step 2: Arranging the Units
- Arranging 5 units around a table can be done in (5-1)! = 4! = 24 ways.
Step 3: Arranging the Ladies within the Block
- The two ladies in the block can be arranged in 2! = 2 ways.
Step 4: Arranging the Third Lady
- The third lady can sit in any of the remaining positions, which is already covered by the arrangement of units.
Step 5: Total Calculation
- Total arrangements = (Ways to choose pairs) × (Arrangement of units) × (Arrangement within block)
\[ \text{Total} = 3 \times 24 \times 2 = 144 \]
Final Answer
The total number of ways for the three ladies and three gents to sit around the table with exactly two ladies sitting together is **144**.
To find the number of ways three ladies (L1, L2, L3) and three gents (G1, G2, G3) can sit around a table such that exactly two ladies sit together, we need to follow specific steps.
Step 1: Treating Two Ladies as One Unit
- Choose any two ladies to sit together. The combinations are:
- (L1, L2), (L1, L3), (L2, L3) → 3 ways.
- Treat the chosen pair as a single unit or block (LL). Now we have:
- 1 block (LL), 1 lady (L), and 3 gents (G1, G2, G3). Total = 1 + 1 + 3 = 5 units.
Step 2: Arranging the Units
- Arranging 5 units around a table can be done in (5-1)! = 4! = 24 ways.
Step 3: Arranging the Ladies within the Block
- The two ladies in the block can be arranged in 2! = 2 ways.
Step 4: Arranging the Third Lady
- The third lady can sit in any of the remaining positions, which is already covered by the arrangement of units.
Step 5: Total Calculation
- Total arrangements = (Ways to choose pairs) × (Arrangement of units) × (Arrangement within block)
\[ \text{Total} = 3 \times 24 \times 2 = 144 \]
Final Answer
The total number of ways for the three ladies and three gents to sit around the table with exactly two ladies sitting together is **144**.
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Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is?
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Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is?.
Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three ladies and three gents can be at around table so that any two and only two of ladies sit together the number of ways is?.
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