Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 whit

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Gujarati

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6.Permutation and Combination

hard

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is $........$

$6860$

$6859$

$6850$

$6589$

(JEE MAIN-2023)

Solution

$7$ Red apple(RA),$5$ white apple $(WA)$,$8$ oranges$(O)$,$5$ fruits to be selected (Note:- fruits taken different) Possible selections :- $(2 O , 1 RA , 2 WA )$ or $(2 O$, $2 RA , 1 WA )$ or ( $3 O , 1 RA , 1 WA )$

$\Rightarrow{ }^8 C _2{ }^7 C _1{ }^5 C _2+{ }^8 C _2{ }^7 C _2{ }^5 C _1+{ }^8 C _3{ }^7 C _1{ }^5 C _1$

$\Rightarrow 1960+2940+1960$

$\Rightarrow 6860$

Standard 11

Mathematics

Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$

medium

(IIT-2017)

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normal

(IIT-2000)

There are $12$ volleyball players in all in a college, out of which a team of $9$ players is to be formed. If the captain always remains the same, then in how many ways can the team be formed

easy

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easy

$^n{C_r}{ + ^n}{C_{r – 1}}$ is equal to