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Number of six-digit numbers such that any digit that appears in the number appears at least twice, where the digits of each number are from the set `{1, 2, 3, 4, 5},` is (Example 225252 is valid but 222133 is not valid)A. `1500`B. `1850`C. `1405`D. `1205` |
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Answer» Correct Answer - C `(c )` Case I : All six digits alike i.e. `111111`, `222222`……..etc. `=5` ways Case II : `2` alike `+2` other alike. Select any three in `"^(5)C_(3)` ways (i.e.`1,2,3` and take `11,22,33`) For each such selections number of ways `=(6!)/(2!2!2!)=90` `implies` Total `=10xx90=900` Case III : `2` alike `+4` other alike i.e. `11 2222` or `22 11 11` etc. Number of ways selecting `2` digits `=("^(5)C_(2))(2)=20` For each selections number of ways `=(6!)/(2!4!)=15` `implies` Total `=20xx15=300` Case IV : `3` alike `+3` other alike Select any two in `"^(5)C_(2)=10` ways For each selection number of ways `=(6!)/(3!*3!)=20` `implies` Total `=10xx20=200` Hence total`=5+900+300+200=1405` |
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