COUNT THE POSSIBILITIES
High-Card Probability and the Category Check
High card occupies 16,440 of the 22,100 standard three-card hands, approximately 74.3891 percent. It is the remainder after trail, pure sequence, sequence, colour and pair receive their own categories. Summing all six category counts gives the original sample space, providing a useful arithmetic and classification check for the entire probability table.
The rule, precisely
High card occupies 16,440 of the 22,100 standard three-card hands, approximately 74.3891 percent. It is the remainder after trail, pure sequence, sequence, colour and pair receive their own categories. Summing all six category counts gives the original sample space, providing a useful arithmetic and classification check for the entire probability table.
The five other category counts are 52, 48, 720, 1,096 and 3,744. Their sum is 5,660. Subtracting that sum from 22,100 gives 16,440. This residual method works because the categories are mutually exclusive and together cover all hands under the study profile.
An independent classification routine can verify the same result by visiting every three-card combination. For each hand, test repeated ranks, sequence membership and suit pattern, then increment exactly one category counter.
Original Practice Hand
Classify the displayed high-card example, then connect its category to the full enumeration. This links a single hand with the arithmetic behind the category table. Keep the stated worksheet assumptions beside the cards and record the specific property that supports the worked explanation.
Work through the cards
Verify that all three ranks differ and the suits are mixed. Check that A-J-8 is outside the declared sequence triples. Assign the hand to high card. It contributes one item to the residual category. Keep the stated worksheet assumptions beside the cards and record the specific property that supports the worked explanation.
Verify that all three ranks differ and the suits are mixed. Check that A-J-8 is outside the declared sequence triples.
Assign the hand to high card. It contributes one item to the residual category.
Add the other category counts: 52 + 48 + 720 + 1,096 + 3,744 = 5,660.
Subtract 5,660 from 22,100 and verify that adding the resulting 16,440 restores the full denominator.
Result and next variation
The high-card count is 16,440. The six category totals sum exactly to 22,100, and their exact fractions sum to one. Keep the stated worksheet assumptions beside the cards and record the specific property that supports the worked explanation. The following steps make that reasoning reproducible for independent study.
Round each category percentage to four decimal places and add the displayed values. A tiny rounding difference can appear in a displayed percentage sum, so use the exact counts for the definitive partition check.
Sources and editorial method
The source note separates published card-rule conventions from the Hub’s own worked examples and mathematical enumeration. Read the declared profile when a convention affects a comparison. Exact cards and calculation steps make the original analysis reproducible, while the editorial policy explains authorship, review scope and the method used to maintain each learning page.
Study convention: The Hub study profile uses one 52-card pack, six standard categories and suit-neutral comparisons. It places A-K-Q above A-2-3, then K-Q-J down to 4-3-2. Practice points count complete explanations within the current page. Read the named convention beside every worked comparison and keep rummy transfer lessons within their own rule system.
Rule reference: Teen Patti rule reference. The Teen Patti reference uses an A-2-3-first ordering; this worksheet uses the declared Hub order. Analysis: fixed teaching deal GH-18; review method by Teen Patti Games Hub Editorial Desk.