Question

Six cards are drawn one by one from a set of unlimited number of cards, each card is marked with numbers $-1,0,$ or $1$. Number of different ways in which they can be drawn if the sum of the numbers shown by them vanishes is

• A. $111$
• B. $121$
• C. $141$
• D. none

Answer 1

Answer: (C)

$141$

Given that sum must vanish

$\Rightarrow$ sum $=0$

possibilities : All $0^{\prime }s,{31}^{\prime }s$ $\xi$ $3(-1{)}^{\prime }s,$

${21}^{\prime }s$ $2(-1{)}^{\prime }s$ $\xi$ ${20}^{\prime }s,$

${11}^{\prime }s$ $1(-1{)}^{\prime }s$ ${40}^{\prime }s$

For case $\left(1\right)$ $-$ only $1$ possible way

For case $\left(2\right)$ $-$ $\frac{6!}{3!3!}$ $=20$ ways

For case $\left(3\right)$ $-$ $\frac{6!}{2!2!2!}$ $=90$ ways

For case $\left(4\right)$ $-$ $\frac{6!}{4!}$ $=30$ ways

Total $=1+20+90+30$

$=141.$

Hence, the answer is $141.$