Problem Solving

AAPL wrote:Manhattan Prep

Gita, Hussain, Inge, Jeong, Karen, and Leila are seated in a row of six chairs. How many seating arrangements are possible if Gita cannot sit next to Inge and Jeong must sit next to Leila?

A. 288
B. 240
C. 144
D. 120
E. 96

OA C

So, there are 6 friends. We want that J and L sit next to each other; consider them as one unit. Thus, there are 5 friends, now.

The number of ways 5 friends can sit together in a row = 5!. However, J and L can exchange their seats and still be next to each other.

Thus, the no. of ways 6 friends be seated in a row such that J and L sit next to each other = 2*5! = 240

The 240 ways include the no. of ways in which G and I sit together; we must exclude them. Let's also consider them as one unit. So, there are 2 units that are grouped; G & I and J & L. And there are 4 units.

No. of ways G & I and J & L sit together = 2*2*4! = 96

Thus, the required no. of ways = 240 - 96 = 144

The correct answer: C

Hope this helps!

-Jay
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AAPL wrote:Manhattan Prep

Gita, Hussain, Inge, Jeong, Karen, and Leila are seated in a row of six chairs. How many seating arrangements are possible if Gita cannot sit next to Inge and Jeong must sit next to Leila?

A. 288
B. 240
C. 144
D. 120
E. 96

OA C

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Let's first assume that the first requirement (Gita cannot sit next to Inge) does not exist but the second (Jeong must sit next to Leila) does. Then we can assume Jeong and Leila as "1 person". So the number of ways to arrange "5 people" is 5! = 120. However, within that "1 person," the sitting arrangement of Jeong and Leila can be either JL or LJ. Therefore we have to multiply 120 by 2 and obtain 240 arrangements.

Of course, the 240 arrangements we've obtained above omits the first requirement. So now let's consider that requirement. However, more precisely, let's consider the opposite of that requirement; that is, let's consider that Gita must sit next to Inge. If that is the case, like before, we can consider Gita and Inge as "1 person". So the number of ways to arrange "4 people" is 4! = 24. However, these "4 people" consist of two "1 person" entities: Gita and Inge, and Jeong and Leila. Therefore we have to multiply 24 by 2 (for GI and IG) and again by 2 (for JL and LJ) to obtain 96 arrangements.

Since we are really looking for the number of arrangements where Gita and Inge cannot sit next to each other, then the number of such arrangements is 240 - 96 = 144.

Answer: C