Class 11

Math

Algebra

Permutations and Combinations

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

- $111$
- $121$
- $141$
- none

Given that sum must vanish

$⇒$ sum $=0$

possibilities : All $0_{′}s,31_{′}s$ $ξ$ $3(−1)_{′}s,$

$21_{′}s$ $2(−1)_{′}s$ $ξ$ $20_{′}s,$

$11_{′}s$ $1(−1)_{′}s$ $40_{′}s$

For case $(1)$ $−$ only $1$ possible way

For case $(2)$ $−$ $3!3!6! $ $=20$ ways

For case $(3)$ $−$ $2!2!2!6! $ $=90$ ways

For case $(4)$ $−$ $4!6! $ $=30$ ways

Total $=1+20+90+30$

$=141.$

Hence, the answer is $141.$