A quadruple (a, b, c, d) of distinct integers is said to be balanced if a + c = b + d. Let S be any

Question

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a + c = b + d. Let S be any set of quadruples (a, b, c, d) where 1 \le a < b < d < c \le 20 and where the cardinality of S is 4411. Find the least number of balanced quadruples in S.

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"type":"text","text":" = "},{"attrs":{"latex":"b"},"type":"math_inline"},{"text":" + ","type":"text"},{"attrs":{"latex":"d"},"type":"math_inline"},{"text":"             a<b<d<c","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 37                (20, 17), (18, 19)                          →       2      →      1 way","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 3:","type":"text","marks":[{"type":"bold"}]},{"type":"text","text":" a + "},{"attrs":{"latex":"c"},"type":"math_inline"},{"type":"text","text":" = 36                (20, 16), (19, 17)                          →       2      →      1 way"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 4:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 35                (20, 15), (19, 16), (18, 17)                →       3      →      3C","type":"text"}]},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 5:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 34                (20, 14), (19, 15), (18, 16)                →       3      →      3C","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 6:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 33                (20, 13), (19, 14)…(18, 15), (17, 16)       →       4      →      4C","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 7:","type":"text","marks":[{"type":"bold"}]},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"type":"text","text":" = 32                (20, 12), (19, 13)…(17, 15)                 →       4"}],"type":"paragraph"},{"content":[{"text":"Step 8:","type":"text","marks":[{"type":"bold"}]},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 31                (20, 11), (16, 15)                          →       5","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 9:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"type":"text","text":" = 30                (20, 10), (16, 14)                          →       5"}],"type":"paragraph"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 10:"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 29 → 6            (20, 9)… (15, 14)                           →       6","type":"text"}]},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 11:"},{"text":" a + ","type":"text"},{"type":"math_inline","attrs":{"latex":"c"}},{"text":" = 28 → 6            (20, 8)\dots.. (15, 13)                       →       6","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 12:"},{"type":"text","text":" a + "},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 27 → 7            (20, 7)\dots. (14, 13)                        →       7","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 13:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 26 → 7","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 14:","type":"text"},{"text":" a + ","type":"text"},{"type":"math_inline","attrs":{"latex":"c"}},{"text":" = 25 → 8","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 15:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"type":"text","text":" = 24 → 8"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 16:","type":"text"},{"type":"text","text":" a + "},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 23 → 9","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 17:","type":"text"},{"type":"text","text":" a + "},{"type":"math_inline","attrs":{"latex":"c"}},{"text":" = 22 → 9","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 18:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 21 → 10           (20, 1)…. (11, 10)","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 19:","type":"text"},{"text":" a + ","type":"text"},{"type":"math_inline","attrs":{"latex":"c"}},{"type":"text","text":" = 20 → 9            (19, 1), (18, 2) \dots.. (11, 9)"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 20:","type":"text"},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 19 → 9            (18, 1)…. (11, 9)","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"text":"Step 21:","type":"text","marks":[{"type":"bold"}]},{"text":" a + ","type":"text"},{"attrs":{"latex":"c"},"type":"math_inline"},{"text":" = 18 → 8            (17, 1), (10, 8)","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 22:","type":"text"},{"text":" a+c=5→2                   (4, 1) (3, 2)","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 23:"},{"text":" (                  )","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 24:","type":"text"},{"text":" Total balanced quadruple = 4 2 C2 + 3C2 \dots 9C2 + 10C2","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 25:"},{"text":" = 4 10C3 + 10C2","type":"text"}]},{"content":[{"text":"Step 26:","type":"text","marks":[{"type":"bold"}]},{"text":" 4 \times 10 \times 9 \times 8","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 27:"},{"text":" =                  + 45 = 525","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"text":"Step 28:","type":"text","marks":[{"type":"bold"}]},{"text":" 4 – 4411 = 434","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 29:","type":"text"},{"type":"text","text":" For least balanced = 525 – 434 = 91"}],"type":"paragraph"}],"type":"doc"}

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