These are three pots and four coins- All these coins are to be distributed into these pots where any pot can contain any number of coins- array - c - c - Whine - Column 1 - - Column2 WhlineA - The number of - ways in - which - all these - coins can we - P - 3 l- distributed if all coins are identical but all pots are different - - llihlineB - The number of - ways - in - which all these coins can we - Q - 6 - distributed if - all coins are different - but - all pots are identical - - MhlineC - The number - of - ways - all these coins - can - be - distributed - such - R - 15 - identical- - - lWhlineD - The number - of - ways - all these coins can be distributed - S - 14 1end array A- A - R - B - S - C - Q - D - PB- A - P - B - Q - C - S - D - RC- A - Q - B - R - C - P - D - SD- A - S - B - P - C - R - D - Q

The correct option is A AR, BS, CQ, DP A → R,B → S,C → Q,D → P. (A)Required number of ways= ^6C_2 (B)Since pots are identical,there will be 4 cases (4,0,0),(3 ...

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Byju's Answer

Standard XII

Physics

New Capacitance in Dielectric

These are thr...

Question

These are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins.

$\begin{matrix} C o l u m n 1 & C o l u m n 2 (A) & T h e n u m b e r o f w a y s i n w h i c h a l l t h e s e c o i n s c a n b e & (P) & 3 d i s t r i b u t e d i f a l l c o i n s a r e i d e n t i c a l b u t a l l p o t s a r e d i f f e r e n t (B) & T h e n u m b e r o f w a y s i n w h i c h a l l t h e s e c o i n s c a n b e & (Q) & 6 d i s t r i b u t e d i f a l l c o i n s a r e d i f f e r e n t b u t a l l p o t s a r e i d e n t i c a l (C) & T h e n u m b e r o f w a y s a l l t h e s e c o i n s c a n b e d i s t r i b u t e d s u c h & (R) & 15 t h a t n o p o t i s e m p t y i f a l l c o i n s a r e d i f f e r e n t b u t a l l p o t s a r e i d e n t i c a l . (D) & T h e n u m b e r o f w a y s a l l t h e s e c o i n s c a n b e d i s t r i b u t e d & (S) & 14 s u c h t h a t n o p o t i s e m p t y i s a l l c o i n s a r e i d e n t i c a l b u t a l l p o t s a r e d i f f e r e n t . \end{matrix}$

A P, B Q, C S, D R

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AR, BS, CQ, DP

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AQ, BR, CP, DS

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AS, BP, CR, DQ

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Solution

The correct option is B
AR, BS, CQ, DP

A $\to$ R,B $\to$ S,C $\to$ Q,D $\to$ P.
(A)Required number of ways= $^{6} C_{2}$

(B)Since pots are identical,there will be 4 cases (4,0,0),(3,1,0),(2,2,0),(2,1,1) but all the coins are different hence selection of coins matters.

For the first case no. of selections = $^{4} C_{4}$ =1

For the second case No.of selection= $^{4} C_{3} \times^{1} C_{1} = 4$

For the third case No.of selection= $\frac{^{4} C_{2} \times^{2} C_{2}}{2!} = 3$

For the third case No.of selection= $\frac{^{4} C_{2} \times^{2} C_{1} \times^{1} C_{1}}{2!} = 6$

(C) Since no box is empty and all pots are identical so the possible case is (1,1,2). But since all the coins are different, the 2 balls can be selected in $^{4} C_{2}$ ways and rest can be put in $\frac{^{2} C_{1} \times^{1} C_{1}}{2!}$

Required number of distributions = $\frac{^{4} C_{2} \times^{2} C_{1} \times^{1} C_{1}}{2!} = 6$

(D) Since no pot is empty and all coins are identical the possible case is (1,1,2). But since all three pots are different hence. a pot (which contains 2 coins together) can be selected in $^{3} C_{1}$ ways. Hence,the required number of distributions= $^{3} C_{1} \times 1 = 3$

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These are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins.
$\begin{matrix} C o l u m n - I & C o l u m n - I I (A) & T h e n u m b e r o f w a y s i n w h i c h a l l t h e s e & (P) & 3 c o i n s c a n b e d i s t r i b u t e d i f a l l c o i n s a r e i d e n t i c a l b u t a l l p o t s a r e d i f f e r e n t (B) & T h e n u m b e r o f w a y s i n w h i c h a l l & (Q) & 23 t h e s e c o i n s c a n b e d i s t r b u t e d i f a l l c o i n s a r e d i f f e r e n t b u t a l l p o t s a r e i d e n t i c a l (C) & T h e n u m b e r o f w a y s a l l t h e s e c o i n s c a n & (R) & 15 b e d i s t r i b u t e d s u c h t h a t n o p o t i s e m p t y i f a l l c o i n s a r e d i f f e r e n t b u t a l l p o t s a r e i d e n t i c a l (D) & T h e n u m b e r o f w a y s a l l t h e s e c o i n s c a n b e & (S) & 12 d i s t r i b u t e d s u c h t h a t n o p o t i s e m p t y i f a l l c o i n s a r e i d e n t i c a l b u t a l l p o t s a r e d i f f e r e n t \end{matrix}$

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