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NVAMediumJEE 2023Combinations (C(n,r))

JEE Mathematics 2023 Question with Solution

Suppose Anil's mother wants to give 55 whole fruits to Anil from a basket of 77 red apples, 55 white apples, and 88 oranges. If in the selected 55 fruits, at least 22 oranges, at least one red apple, and at least one white apple must be given, then the number of ways Anil’s mother can offer 55 fruits to Anil is:

Answer

Correct answer:6860

Step-by-step solution

Standard Method

Given: There are 77 red apples, 55 white apples, and 88 oranges. Exactly 55 fruits are to be selected such that there are at least 22 oranges, at least one red apple, and at least one white apple.

Find: The number of ways to select the fruits.

Possible selections are:

  1. 22 oranges, 11 red apple, 22 white apples
  2. 22 oranges, 22 red apples, 11 white apple
  3. 33 oranges, 11 red apple, 11 white apple

The total number of ways for each case is

(82)⋅(71)⋅(52)=1960\binom{8}{2} \cdot \binom{7}{1} \cdot \binom{5}{2} = 1960 (82)⋅(72)⋅(51)=2940\binom{8}{2} \cdot \binom{7}{2} \cdot \binom{5}{1} = 2940 (83)⋅(71)⋅(51)=1960\binom{8}{3} \cdot \binom{7}{1} \cdot \binom{5}{1} = 1960

Adding them,

1960+2940+1960=68601960 + 2940 + 1960 = 6860

Therefore, the required number of ways is 68606860.

Casewise Counting

Given: Fruits are distinct within each category, and 55 fruits must be chosen with at least 22 oranges, at least one red apple, and at least one white apple.

Find: The total valid selections.

Since at least one red apple and at least one white apple are required, after choosing at least 22 oranges, the only possible distributions of 55 fruits are:

  • (2,1,2)(2,1,2) = oranges, red apples, white apples
  • (2,2,1)(2,2,1)
  • (3,1,1)(3,1,1)

Now count each case:

(2,1,2):(82)(71)(52)=1960(2,2,1):(82)(72)(51)=2940(3,1,1):(83)(71)(51)=1960\begin{aligned} (2,1,2) &: \binom{8}{2}\binom{7}{1}\binom{5}{2} = 1960 \\ (2,2,1) &: \binom{8}{2}\binom{7}{2}\binom{5}{1} = 2940 \\ (3,1,1) &: \binom{8}{3}\binom{7}{1}\binom{5}{1} = 1960 \end{aligned}

Therefore,

1960+2940+1960=68601960 + 2940 + 1960 = 6860

So the answer is 68606860.

Common mistakes

  • Counting impossible cases such as (4,1,0)(4,1,0) or (2,0,3)(2,0,3) is wrong because at least one red apple and at least one white apple are both compulsory. First enforce all conditions before forming cases.

  • Treating fruits of the same type as identical is wrong. The solution uses combinations because the 77 red apples, 55 white apples, and 88 oranges are chosen from distinct fruits in the basket.

  • Using permutations instead of combinations is incorrect because the order in which the 55 fruits are offered does not matter. Choose fruits by (nr)\binom{n}{r}, not by arrangements.

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