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Permutations and Combinations question

2023 · 24 Jan · Shift 1 · Q41
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Permutations and Combinations question

2023 · 24 Jan · Shift 1 · Q41

JEE MainMathematicsPermutations and CombinationsNumerical+4 / −1
A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is ‾\underline{\hspace{2cm}}​
Numerical answer
View written solutionFree

Correct answer: 546

  1. Understand the data
  • Total courses available =12=12=12
  • Language courses =5=5=5
  • Non-language courses =12−5=7=12-5=7=12−5=7
  • He has to choose exactly 555 courses.
  • Condition: at most 2 language courses

So the number of language courses chosen can be:

  • 000
  • 111
  • 222

We count each case separately.

  1. Case 1: Choose 0 language courses

Then all 555 courses must be chosen from the 777 non-language courses.

(50)(75)=1⋅21=21\binom{5}{0}\binom{7}{5}=1\cdot 21=21(05​)(57​)=1⋅21=21

  1. Case 2: Choose 1 language course
  • Choose 111 from the 555 language courses
  • Choose remaining 444 from the 777 non-language courses

(51)(74)=5⋅35=175\binom{5}{1}\binom{7}{4}=5\cdot 35=175(15​)(47​)=5⋅35=175

  1. Case 3: Choose 2 language courses
  • Choose 222 from the 555 language courses
  • Choose remaining 333 from the 777 non-language courses

(52)(73)=10⋅35=350\binom{5}{2}\binom{7}{3}=10\cdot 35=350(25​)(37​)=10⋅35=350

  1. Total number of ways

21+175+350=54621+175+350=54621+175+350=546

So, the required number of ways is

546\boxed{546}546​

  1. Comparison with stored correct answer
  • Derived answer: 546546546
  • Stored correct answer: 546546546

They match.

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