Question

Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 and Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss.

Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?

• A. Spotted came in first
• B. Red finished last
• C. Black came in second
• D. White came in second
• E. There was one horse between Black and White

Hint:

No profit and no loss means the money Raju gets back equals the Rs. 6000 he staked. Use this to find where Red, White and Black can finish, then put Grey fourth.

Answer 1

The Correct Option is D

Step 1: Turn the rules into returns:
The winning horse returns 4 times the bet, the second horse 3 times, the third horse returns the bet (1 time), and the 4th and 5th horses return 0. Raju staked \(3000+2000+1000=6000\), so he must get back exactly Rs. 6000.

Step 2: Write the condition:
Let \(m_R, m_W, m_B\) be the multipliers earned by Red, White and Black (each is 4, 3, 1 or 0, and different positions give different multipliers, except that two horses can both get 0). Then
\(3000\,m_R+2000\,m_W+1000\,m_B=6000 \;\Rightarrow\; 3m_R+2m_W+m_B=6\).

Step 3: Find all possible results for Red, White and Black:
If \(m_R=4\) or \(3\), then \(3m_R\ge 9>6\), which is impossible. So \(m_R\) is 1 or 0.
If \(m_R=1\) (Red third): \(2m_W+m_B=3\), which gives \(m_W=0,\ m_B=3\). So Black is second and White is 4th or 5th.
If \(m_R=0\) (Red 4th or 5th): \(2m_W+m_B=6\), which gives \((m_W,m_B)=(3,0)\) or \((1,4)\).
\(\bullet\) \((3,0)\): White is second, Black is 4th or 5th, and Grey and Spotted take 1st and 3rd.
\(\bullet\) \((1,4)\): Black is first, White is third, and Grey and Spotted take 2nd and the remaining 4th/5th place.

Step 4: Now use "Grey came in fourth":
In the case with White second, Grey must be 1st or 3rd, so Grey cannot be fourth. This case is out.
Black first, White third: Grey fourth means Spotted is second and Red is fifth. Order: Black, Spotted, White, Grey, Red.
Red third, Black second: Grey fourth means Spotted is first and White is fifth. Order: Spotted, Black, Red, Grey, White.

Step 5: Test the options on these two orders:
(1) Spotted came in first: true in the second order.
(2) Red finished last: true in the first order.
(4) White came in second: White is third or fifth, so this is never true.
(3) Black came in second: true in the second order.
(5) One horse between Black and White: true in the first order (Black 1st, Spotted 2nd, White 3rd).

Final Answer:
The correct option is (4): White came in second cannot be true.