Find number of seven-digit number in the form of `a b c d efg(g,f,e ,` tc. Are digits at units, tens hundreds place etc.) `where a<< b<< c<< d>>e>>f>>

Monjur Alahi

Find number of seven-digit number in the form of `a b c d efg(g,f,e ,` tc. Are digits at units, tens hundreds place etc.) `where a<< b<< c<< d>>e>>f>>g dot`
A. `1980`
B. `1116`
C. `1560`
D. `1476`


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Salim Ahmed Kawsar

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Correct Answer - C
`(c )` Case (i) : zero not taken.
Now we have to select seven digits from `1,2,3,4,5,6,7,8,9` so ways are `"^(9)C_(7)`
From `7` digit select the largest digits as `d` from remaining `6`, we can select three digits `a`, `b`, `c` in `.^(6)C_(3)` ways.
Hence number of such numbers are `=.(9)C_(7)*^(6)C_(3)`
Case (ii) : zero taken then
`0` must be at last place, then number of such numbers are `.^(9)C_(6)*^(5)C_(3)`
So total ways `.^(9)C_(7)*^(6)C_(3)+^(9)C_(6)*^(5)C_(3)=^(9)C_(2)*^(6)C_(3)+^(9)C_(3)*^(5)C_(3)`
`=1560`

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