JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Permutations and Combinations

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There are 5 students in class 10,6 students in class 11 and 8 students in class 12 . If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100 k , then k is equal to ____

Answer: 238

Solution

Let

x, y, z

be the numbers chosen from classes 10, 11 and 12, so

x + y + z = 10

with

x, y, z \geq 2

x \leq 5

y \leq 6

z \leq 8

and

x + y \leq 5

. From

x \geq 2

and

y \geq 2

x + y \geq 4

, so

x + y

is either

4

5

.
For

x + y = 4

the only split is

(2, 2)

, with

z = 6

; for

x + y = 5

the splits are

(2, 3)

and

(3, 2)

, each with

z = 5

. The count is therefore

(2 5) (2 6) (6 8) + (2 5) (3 6) (5 8) + (3 5) (2 6) (5 8)

= 10 \cdot 15 \cdot 28 + 10 \cdot 20 \cdot 56 + 10 \cdot 15 \cdot 56 = 4200 + 11200 + 8400 = 23800

Since

23800 = 100 \cdot 238

, the value of

k

238

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Integer answer

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JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Permutations and Combinations

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