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6.Permutation and Combination
hard
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 $100 \mathrm{k}$, then $\mathrm{k}$ is equal to $......$
A
$240$
B
$245$
C
$270$
D
$238$
(JEE MAIN-2021)
Solution
| Class | $10^{\text {th }}$ | $11^{\text {th }}$ | $12^{\text {th }}$ | |
| Total student | $5$ | $6$ | $8$ | |
| $2$ | $3$ | $5$ | $\Rightarrow{ }^{5} C_{2} \times{ }^{6} \mathrm{C}_{3} \times{ }^{8} \mathrm{C}_{5}$ | |
| Number of selection | $2$ | $2$ | $6$ | $\Rightarrow{ }^{5} \mathrm{C}_{2} \times{ }^{6} \mathrm{C}_{3} \times{ }^{8} \mathrm{C}_{6}$ |
| $3$ | $2$ | $5$ | $\Rightarrow{ }^{5} \mathrm{C}_{3} \times{ }^{6} \mathrm{C}_{2} \times{ }^{8} \mathrm{C}_{5}$ |
$\Rightarrow$ Total number of ways $=23800$
According to question
$100 \mathrm{~K}=23800$
$\Rightarrow \mathrm{K}=238$
Standard 11
Mathematics
