Let H, D, S, C denote the number of hearts, diamonds, spades and clubs respectively.
- Red cards (hearts + diamonds): \(H + D = 8\)
- Black cards (spades + clubs): \(S + C = 7\)
- Diamonds = 2 × spades: \(D = 2S\)
- Hearts = diamonds: \(H = D = 2S\)
From the red-card equation: \(2S + 2S = 8 \Rightarrow S = 2\); hence \(D = 4, H = 4\).
From the black-card equation: \(C = 7 - S = 7 - 2 = \mathbf{5}\).
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On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path ?
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