From a standard deck of 52 cards, a 7-card hand is dealt. In how many distinct 7-card hands will there be exactly 2 spades and exactly 3 diamonds (with the remaining 2 cards coming from the other suits)?

Introduction / Context:
This question is a conditional card combinations problem. We must count 7 card hands with a specified composition of suits: exactly 2 spades, exactly 3 diamonds, and the remaining 2 cards from the other suits (hearts and clubs). These partitioned counting problems are very common in probability and combinatorics involving card games and suit constraints.

Given Data / Assumptions:

• Standard deck: 52 cards, 13 in each suit (spades, hearts, diamonds, clubs).
• We deal a hand of 7 cards.
• Exactly 2 cards must be spades.
• Exactly 3 cards must be diamonds.
• The remaining 2 cards must come from the other 26 cards (hearts and clubs).
• Order of cards in the hand does not matter; only the set of cards matters.

Concept / Approach:
We choose cards suit by suit using combinations:

• Choose 2 spades out of 13 spades: 13C2.
• Choose 3 diamonds out of 13 diamonds: 13C3.
• Choose 2 remaining cards from the 26 non spade, non diamond cards (hearts and clubs): 26C2.

Since these choices are independent and define different subsets, we multiply the three combination counts to get the total number of valid 7 card hands.

Step-by-Step Solution:
Step 1: Count ways to choose spades: 13C2. Step 2: Count ways to choose diamonds: 13C3. Step 3: Cards not spades or diamonds: 52 - 13 - 13 = 26. Step 4: Count ways to choose remaining 2 cards from these 26: 26C2. Step 5: Multiply all three: total hands = 13C2 * 13C3 * 26C2. Step 6: Compute: 13C2 = 78, 13C3 = 286, 26C2 = 325. Step 7: Multiply: 78 * 286 * 325 = 7250100.

Verification / Alternative check:
We can verify intermediate values: 13C2 = 13 * 12 / 2 = 78, and 13C3 = (13 * 12 * 11) / (3 * 2 * 1) = 286. For 26C2, (26 * 25) / 2 = 325. These are standard combination values. Multiplying 78 * 286 gives 22308, and 22308 * 325 equals 7250100, which matches the given correct option. This confirms that all arithmetic steps are consistent.

Why Other Options Are Wrong:
7690030 and 7250000: These are large numbers, but they do not equal the exact product of the correct combination factors. 3454290: Roughly half the true value, suggesting a missing factor or incomplete multiplication. Only 7250100 matches 13C2 * 13C3 * 26C2.

Common Pitfalls:
One common mistake is to forget that the remaining 2 cards must come from the suits other than spades and diamonds, accidentally counting hands with more than 2 spades or more than 3 diamonds. Another is to treat the 7 cards as ordered and use permutations, which overcounts the number of distinct hands. Carefully partitioning by suit and using combinations for each subset avoids these issues.

Final Answer:
The number of 7 card hands with exactly 2 spades, exactly 3 diamonds, and the remaining 2 cards from hearts or clubs is 7250100.