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Combinations are used when order does not matter, so rearranging items does not create a new outcome.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do you calculate permutations of r items chosen from n items?",{"text":68,"@type":64},"Use the permutations formula nPr based on factorial terms: nPr = n! / (n − r)!. The guide emphasizes that sequence matters.",{"name":70,"@type":61,"acceptedAnswer":71},"How should you count passwords using combinations and permutations?",{"text":72,"@type":64},"Determine the character set size per slot (digits, letters, vowels, etc.), the password length (number of slots), and whether repetition is allowed. 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These are counting methods in math that help us answer questions like:  \n􀀷 “How many ways can I arrange or choose items? ˮ They both deal with selecting or arranging items—  \nbut the difference is whether order matters or not.  \n􀀐 Permutations  \nOrder matters  \nIf you're arranging items in a specific sequence, use permutations.  \nFormula:  \nn!  \nnPr = ~~ ~~(n − r)!  \nWhere:  \n  n  = total number of items  \n  r  = number of items being arranged  \n􀃮 Combinations and Permutations | by Lazizbek 1  \n  !  (factorial) means: 5! 􀗿 5􀘃4􀘃3􀘃2􀘃1  \n􀀐 Key idea􀗤 Changing the order creates a new result.✅ Example:  \nHow many ways can you arrange 3 out of 5 books on a shelf?  \n 5!   \n5P3 = (5 − 3)! = 5 × 4 × 3 = 60 ways  \n􀀞 Combinations Order doesnʼt matter  \nIf you're just selecting items without caring about order, use combinations.  \nFormula:  \nn!  \nnCr = ~~ ~~  \nr!(n − r)!  \nWhere:  \n  n  = total number of items  \n  r  = number of items chosen  \n􀀐 Key idea􀗤 Changing the order does not create a new result.✅ Example:  \nHow many ways can you choose 3 students out of 5?  \n 5!   120   \n5C3 = 3!(5 − 3)! = 6 × 2 = 10 ways  \n􀃌 Combinations vs Permutations  \n􀃮 Combinations and Permutations | by Lazizbek 2  \n\n| Feature | Permutations | Combinations |\n| --- | --- | --- |\n| Order? | ✅ Yes, order matters | ❌ No, order doesn't matter |\n| Formula | nPr = n! / (n − r)! | nCr = n! / [r!(n − r)!] |\n| Usually more? | Yes | No |\n\n\n| ✍ Practice Time!\u003Cbr>Try solving these without a calculator:\u003Cbr>􀕒 􀕢 How many ways can you arrange 4 out of 7 letters?\u003Cbr>􀕓 􀕢 How many ways can you choose 2 fruits from a basket of 6?\u003Cbr>􀕔 􀕢 A team of 3 is selected from 10 players. How many combinations are there?􀕕 􀕢 How many permutations can be made from the letters A, B, C, D?\u003Cbr>􀕖 􀕢 Choose 5 books from a shelf of 9 (order does not matter) . |\n| --- |\n| 􀂺 Tip:\u003Cbr>Remember the difference with this trick: Permutation → Position matters.\u003Cbr>Combination → just Choosing. |\n| 􀀐 How to Solve Password Combination Problems |\n\nWhen faced with a password creation problem, always ask yourself these key questions:  \n􀃅 What characters are being used?  \nIs the password made from:  \n 􀀐 Digits 􀖨 0􀗂9􀖩 ?  \n 􀀎 Letters 􀖨 A􀗂Z􀖩 ?  \n 􀀑 A mix of both?  \n➡ Count the total options per character  \nDigits 􀜟 10 options  \nLetters 􀜟 26 options  \n􀃮 Combinations and Permutations | by Lazizbek 3  \nVowels 􀜟 5 options Uppercase letters 􀜟 26 Lowercase letters 􀜟 26  \nAlphanumeric 􀜟 36  \n\n| 􀃆 How many characters long is the password? This gives you the number of slots to fill.\u003Cbr>For example:\u003Cbr>􀃿 A 4-digit password 􀜟 4 slots |\n| --- |\n| 􀃇 Is repetition allowed? This changes the entire approach:\u003Cbr>✅ Repetition allowed\u003Cbr> Multiply choices per slot (use powers):\u003Cbr>104 , 263 , etc.\u003Cbr>❌ No repetition\u003Cbr> Use permutations:\u003Cbr>P (n, r) = (n~~ ~~!~~ ~~r)! |\n| 􀃈 Are there any special conditions?\u003Cbr>Sometimes the password must: Start with a specific character Contain a vowel\u003Cbr>Exclude repetition\u003Cbr>Combine letters and digits\u003Cbr>􀂺 Break the problem into steps.\u003Cbr>First satisfy the condition, then count possibilities. |\n\n􀃮 Combinations and Permutations | by Lazizbek 4  \n􀃺 Example 1:  \nHow many 4-digit passwords can be made from digits 0􀗂9, if repetition is allowed?  \n10 options per digit  \n4 digits →  \n10 × 10 × 10 × 10 = 104 = 10,000  \n\n| 􀃺 Example 2:\u003Cbr>How many 4-digit passwords using 0􀗂9, with no digit repeated?\u003Cbr> 10 options for 1st digit\u003Cbr> 9 for 2nd 􀖨 1 used)\u003Cbr> 8 for 3rd\u003Cbr> 7 for 4th\u003Cbr>→\u003Cbr>10 × 9 × 8 × 7 = 5040 |\n| --- |\n| 􀃺 Example 3:\u003Cbr>How many 6-letter passwords using A􀗂Z, no repetition? P (26, 6) = (262!~~ ~~6)! = 26 × 25 × 24 × 23 × 22 × 21 |\n\n􀀙 Final Tip:  \nIf you're mixing letters and digits, handle each slot carefully.  \nExample: A 3-character password like 􀖪 Letter, Digit, Letter] →  \n􀃮 Combinations and Permutations | by Lazizbek 5  \n26 × 10 × 26 = 6760  \n􀀅 More: t.me/ML_notes  \n􀃮 Combinations and Permutations | by L","cbCaifqLZYZjanIx","https://ap.wps.com/l/cbCaifqLZYZjanIx","pdf",25821213,"English","# Combinations and Permutations\n## Permutations: Order Maters\n## Combinations: Order Doesn't Matter\n## Combinations vs Permutations\n## Practice Time\n## Password Combination Problems\n## Key Questions for Password Counting\n## Repetition Rules and Formulas\n## Worked Examples","[{\"question\":\"What is the main difference between permutations and combinations?\",\"answer\":\"Permutations are used when order matters, so changing the sequence creates a new result. Combinations are used when order does not matter, so rearranging items does not create a new outcome.\"},{\"question\":\"How do you calculate permutations of r items chosen from n items?\",\"answer\":\"Use the permutations formula nPr based on factorial terms: nPr = n! / (n − r)!. The guide emphasizes that sequence matters.\"},{\"question\":\"How should you count passwords using combinations and permutations?\",\"answer\":\"Determine the character set size per slot (digits, letters, vowels, etc.), the password length (number of slots), and whether repetition is allowed. Then apply powers for repetition-allowed cases or permutations for no-repetition cases, and handle any special conditions step by step.\"}]","Combinations and Permutations - Count Arrangements and Selections | PDF"]