Question

HEYYY GUYS ✌✌

After a long time , I have a question 4 u ...


PLZ ANS AS SOON AS POSSIBLE....

The answer is already written in the question bt i want to know the logic....

PLZ HELP ME I WANT THIS BEFORE TOMORROW MORNING☺☺

Answer 1

$\bold{\huge{\underline{Solution\:\: \colon}}}$

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$\bold{\underline{\large{Revision \: of\: formula\: \colon}}}$

$\bold{\large{1) \:^nC_{r} \: = \: \dfrac{n!}{r!\:(n\: - \: r)!}}}$

$2) \: \bold{\large{C[(n+r-1),(r-1)] }}$

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$\bold{\large{\underline{Solution\:1\: \colon}}}$

$\bold{\underline{\large{Given\: \colon}}}$

Number of identical balls = 8

Number of boxes in which these balls are distributed = 3

$\bold{\underline{\large{To \: find\: \colon}}}$

No of ways of distribution of balls

$\bold{\underline{\large{Solution\: \colon}}}$

If we distribute 3 balls in 3 distinct boxes it's never become empty.

$\bold{\large{\boxed{B1}}}$ $\bold{\large{\boxed{B2}}}$ $\bold{\large{\boxed{B3}}}$

Now three balls are distributed remain balls = (8-3) = 5 balls

$C[(n+r-1),(r-1)] \\\\ = C[(5+3-1),(3-1)\\\\ = C[(8-1),2] \\\\ = C[(7,2)]$

$^nC_r \: = ^7C_2 \\\\ \implies \dfrac{7!}{2!(7-2)!} \\\\ \implies \dfrac{7\times6 (5!) }{2\times1 (5!)} \\\\ \implies \dfrac{42 \cancel{(5!)}}{2 \cancel{(5!)}} \\\\ \implies \dfrac{42}{2} \\\\ \implies 21\: ways$

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$\bold{\large{\underline{Solution\: 2\: \colon}}}$

$\bold{\large{\underline{Given\: \colon}}}$

Number of apples = 30

Number of people = 5

$\bold{\large{\underline{To \:find\: \colon}}}$

Number of ways of distribution of 30 apples among five people

$\bold{\large{\underline{Solution\: \colon}}}$

Distribution of 30 apples among 5

$C[(30+5-1),(5-1)] \\\\ = C[(35-1),(4) \\\\ = C[34,4]$

$^nC_r = \dfrac{n!}{r!(n-r)!} \\\\ ^{34}C_4 = \dfrac{34!}{4!(30-4)!} \\\\ \implies \dfrac{34!}{4!\: 30!} \\\\ \implies \dfrac{34\times 33 \times 32 \times 31\: \cancel{30!}}{ 4\times 3\times 2 \:\cancel{(30!)}} \\\\ \implies \dfrac{1113024}{24} \\\\ \implies 46736\: ways$

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$\bold{\underline{\large{ Answer\: 1}}}$

21 ways to distribute 8 identical balls in 3 boxes.

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$\bold{\underline{\large{ Answer\: 2}}}$

46,736 ways to distribute 30 apples among 5 people.

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$\bold{\large{Thanks}}$