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question

Agnishom has 10 pockets and 44 coins . He wants to put his coins into his pockets so distributed that each pocket contains a different number of coins

Can he do so?
Generalize the problem,considering p pockets and n coins.

The problem is most interesting when

$n = \frac{(p + 1)(p - 2)}{2}$

Why?

#Combinatorics #JEE

Easy Math Editor

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Comments

He can not do so. Because He can't keep $2$ or more pockets empty (as it is given that each pocket has different number of coins) . And at the least, if we keep one pocket empty, then also minimum numbers of coins required to put in $9$ pockets is $\displaystyle \sum_{i=1}^{9}=45$ .
Minimum number of coins required to put into p pockets (according to the restrictions applied) $=\dfrac{p(p-1)}{2}$

Sandeep Bhardwaj - 6 years, 7 months ago

yes Sir its right , can you answer last part

U Z - 6 years, 7 months ago

Same is the case with the last part. Minimum no. of coins required $=\dfrac{p(p-1)}{2}$ . And as you mentioned there , if no. of coins is $\dfrac{(p+1)(p-2)}{2}$ which is one less than the no. of coins required. I want to say that $\dfrac{p(p-1)}{2}-\dfrac{(p+1)(p-2)}{2}=1$ . So, in this case too, it is not possible to put the coins into the pockets with the restrictions applied. @megh choksi

Sandeep Bhardwaj - 6 years, 7 months ago

@Sandeep Bhardwaj – Yes right , thank you for replying

U Z - 6 years, 7 months ago

1+2+3+4+5+6+7+8+9=45. He will have one empty pocket!

Venture HI - 6 years, 7 months ago

Sorry @Venture HI , I was waiting for your response to the second part . nicely done ,voted up thank you

U Z - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$