A group of 540 persons is to be seated row wise such that the number of persons in each row is 4 less than in the previous row. Which of the following number of rows is not possible?

Correct option is C

​Given:

• Total number of people to be seated = 540.
• The number of people in each row decreases by 4 compared to the previous row.
• We need to find which of the given numbers of rows nnn is not possible.

Formula:

The total number of people is modeled as an arithmetic progression (AP), where:

• First term (aaa) = xxx (number of people in the first row),
• Common difference (ddd) = −4-4−4,
• Number of terms (nnn) = number of rows.

The sum of an AP is given by:

S_n =  n/2 [2a+(n−1)d]

Here:

• $S=540$,
• $a=x$,
• $d=-4$.

Derivation:

Substitute the values into the formula:

540 = n/2 [2x+(n−1)(−4)]

Simplify:

1080=n[2x−4n+4]1080 = n [2x - 4n + 4]$$1080=n(2x-4n+4)$$

Checking Each Option

1. Option 1: $n=5$

​$$1080=5(2x-4(5)+4)$$1080=5(2x−20+4)1080 = 5 (2x - 20 + 4)

$$1080=5(2x-20+4)$$​

This is valid. 5 rows is possible.

1. Option 2: $n=6$

​$$1080=6(2x-4(6)+4)$$

1080=6(2x−24+4)1080 = 6 (2x - 24 + 4)1080=6(2x−24+4)1080=6(2x−20)1080 = 6 (2x - 20)1080=6(2x−20)180=2x−20180 = 2x - 20180=2x−202x=180+20=2002x = 180 + 20 = 2002x=180+20=200x=100x = 100x=100​

This is valid. 6 rows is possible.

1. Option 3: $n=8$

​$$1080=8(2x-4(8)+4)$$1080=8(2x−32+4)1080 = 8 (2x - 32 + 4)

$$1080=8(2x-32+4)$$​

This is not valid because $x$ is not an integer. 8 rows is not possible.

1. Option 4: $n=9$

​$$1080=9(2x-4(9)+4)$$1080=9(2x−36+4)1080 = 9 (2x - 36 + 4)

$$1080=9(2x-36+4)$$​

This is valid. 9 rows is possible.

Final Answer

The number of rows that is not possible is:

Option c: 8 rows.