Question

A player holds 13 cards of four suits, of which seven are black and six are red. There are twice as many diamonds as spades and twice as many hearts as diamonds. How many clubs does he hold?

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

In card problems, always separate suits into black and red first, then apply given ratios—this simplifies equation formation.

Answer 1

The Correct Option is C

Concept: Use suit classification:

• Black cards = Spades + Clubs
• Red cards = Hearts + Diamonds

and form equations based on given ratios. Step 1:Let number of spades = $x$.
Then: \[ \text{Diamonds} = 2x,\quad \text{Hearts} = 2 \times (2x) = 4x \]
Step 2:Use total cards.} \[ x + 2x + 4x + \text{Clubs} = 13 \Rightarrow 7x + \text{Clubs} = 13 \quad \cdots (1) \]
Step 3:Use black cards condition.} \[ \text{Spades} + \text{Clubs} = 7 \Rightarrow x + \text{Clubs} = 7 \quad \cdots (2) \]
Step 4:Solve equations.} From (2): \[ \text{Clubs} = 7 - x \] Substitute into (1): \[ 7x + (7 - x) = 13 \Rightarrow 6x + 7 = 13 \Rightarrow 6x = 6 \Rightarrow x = 1 \]
Step 5:Find number of clubs.} \[ \text{Clubs} = 7 - 1 = 6 \] But rechecking total cards: \[ 1 + 2 + 4 + 6 = 13 \quad \checkmark \] Final Answer: \[ {6} \]