Question

A seven-digit number is in the form of $abcdefg(g,f,e,\dots$ 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 ${}^9{C}_6\dots {}^5{C}_3$
So total ways ${}^9{C}_7\cdot {}^6{C}_3+{}^9{C}_6\cdot {}^5{C}_3={}^9{C}_2\cdot {}^6{C}_3+{}^9{C}_3\cdot {}^5{C}_3$
$=1560$