Question

A seven-digit number is in the form of $a b c d e f g(g, f, e, \ldots$ etc. are digits at units, tens, hundred place ....etc), where $a < b < c < d > e > f > g$ and $a, b, c, e, f, g$ are different digits. The number of such numbers is

• A. 1980
• B. 1116
• C. 1560
• D. 1476

Answer 1

Answer: (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 $d$ and from remaining $6$, we can select three digits $a, b, c$ in ${ }^{6} C _{3}$ ways.
Hence number of such numbers are $={ }^{9} C_{7} \cdot{ }^{6} C_{3}$
Case (ii): zero taken.
0 must be at last place, then number of such numbers are ${ }^{9} C _{6} \ldots{ }^{5} C _{3}$
So total ways ${ }^{9} C_{7} \cdot{ }^{6} C_{3}+{ }^{9} C_{6} \cdot{ }^{5} C_{3}={ }^{9} C_{2} \cdot{ }^{6} C_{3}+{ }^{9} C_{3} \cdot{ }^{5} C_{3}$
$=1560$