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)

Options

A. `1500`
B. `1850`
C. `1405`
D. `1205`

Answer

C

Explanation

`(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`