Why does the nPr OR nCr formula **not** used here?

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3 votes

There are $6$ boxes numbered $1, 2, \dots\dots,6$. Each box is to be filled up either with a red or a green ball in such a way that at least $1$ box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is :

- $18$
- $19$
- $20$
- $21$

1 vote

At least one box has green ball

so green ball in 1, or 12 , 123 or 1234 or 12345 , 123456 total =6 cases

or in 2 , 23 or 234 or 2345 , 23456 total =5 cases

or in 3 , 34 or 345 or 3456 total =4 cases

or in 4 , 45 or 456 total =3 cases

or in 5, 56 total 2 cases or in 6 only total 1 case

so total no of ways = 6+5+4+3+2+1=**21 option D**

1 vote

- If one green bail in a box, then the number of ways $= 6\:(1$ box has the green ball can be any of the $6$ boxes.$)$
- If two green balls in a box, then the number of ways $= 5 \:(2$ boxes have green balls The boxes may be numbered as $12, 23, 34, 45, 56)$
- If three green balls in a box, then the number of ways $= 4\:(3$ boxes $123, 234, 345, 456)$
- If four green balls in a box, then the number of ways $= 3\:(4$ boxes $1234,2345,3456)$
- lf five green balls in a box, then the number of ways $= 2\:(5$ boxes $12345,23456 )$
- If six green balls in a box, then the number of ways $= 1\:(6$ boxes $123456)$

$\therefore$ Total number of ways $= 6 + 5 + 4 + 3 + 2 + 1 = 21.$

So, the correct answer is $(D).$