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一副二色牌共有纸牌22张，其中红，蓝每种颜色各 11 张，编号分别为 0,1 ， $2, \ldots, 10$ ，从这副牌中任取若干张牌，然后按照如下规则计算分值：每张编号为 $k$ 的牌记为 $2^{k}$ 分，若它们的分值之和为2025，就称这些牌为一个＂好＂牌组，则＂好＂牌组的个数为 $\qquad$ ．

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∻Solution by Steps∻ ‖ step 1 ⋮ We need to find the number of subsets of the cards such that the sum of their values equals 2025. Each card with number $ k $ contributes $ 2^k $ points. The values of the cards are $ 2^0, 2^1, 2^2, \ldots, 2^{10} $. ‖ ‖ step 2 ⋮ The total value of all cards is given by the sum $ 2^0 + 2^1 + 2^2 + \ldots + 2^{10} = 2^{11} - 1 = 2047 $. ‖ ‖ step 3 ⋮ To find subsets that sum to 2025, we can calculate $ 2047 - 2025 = 22 $. This means we need to find subsets of cards that sum to 22. ‖ ‖ step 4 ⋮ The value 22 can be represented as a sum of powers of 2. The binary representation of 22 is $ 10110_2 $, which corresponds to $ 2^4 + 2^3 + 2^1 $. Thus, we can use the cards numbered 1, 3, and 4. ‖ ‖ step 5 ⋮ The number of ways to select cards is determined by the combinations of the remaining cards. Since we have 11 cards of each color, we can choose any combination of the remaining cards. The total number of combinations is $ 2^3 = 8 $ (including the empty set). ‖ ∻Answer∻ ⚹ The number of "good" card groups is 8. ⚹ ∻Key Concept∻ ⚹ The problem involves finding subsets of a set that sum to a specific value using powers of 2. ⚹ ∻Explanation∻ ⚹ The solution involves calculating the total value of the cards and determining which combinations can achieve the target sum of 2025. The binary representation helps identify the necessary cards. ⚹◊What is the maximum number of cards you can select from the deck to achieve a sum of $2025$ using the scoring rule $2^{k}$ for card values?⍭ Generate me a similar question◊

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