Question

Using the digits $1,2,3,4,5,6,7$, a number of 4 different digits is formed. Find

Column I		Column II
A	How many numbers are formed?	1	840
B	How many numbers are exactly divisible by 2 ?	2	200
C	How many numbers are exactly divisible by 25 ?	3	360
D	How many of these are exactly divisible by 4 ?	4	40

Match the questions in Column I with Column II and choose the correct option from the codes given below.

• A. A - (1), B -(2), C - (3), D -(4)
• B. A - (3), B -(1), C - (4), D -(2)
• C. A - (1), B -(3), C - (4), D -(2)
• D. A - (4), B -(2), C - (3), D -(1)

Answer 1

Answer: (C)

A. The number of 4 different digits $={ }^7 P_4$
$=\frac{7 !}{(7-4) !}$
$=7 \times 6 \times 5 \times 4=840$
B. The numbers exactly divisible by 2
= Number of ways of filling first 3 places $\times$ Number of ways of filling units's place
$={ }^6 P_3 \times 3$
$=\frac{6 !}{(6-3) !} \times 3=\frac{6 !}{(3 !)} \times 3$
$=6 \times 5 \times 4 \times 3=360$
C. Number of 4-digit numbers divisible by $25=$ Numbers ending with 25 or 75
$=\overset{5\times 4}{\square\square}\,\,\,\,\,\overset{25\,or\,75}{\square\square}$
$=5 \times 4 \times 2=40$
$(\because$ when numbers end with 25 or 75 , the other two places can be filled in 5 and 4 ways)
D. Number of 4-digit numbers divisible by $4=$ Numbers ending with $12,16,24,32,36,64,72,76,52,56$
Now, number ending with $12= \underset{4 \times }{\square} \underset{5 \times }{\square}\underset{1\times }{\square} \underset{1}{\square}=20$
Similarly, numbers ending with other number $(16,24, \ldots)=20$ each
$\therefore$ Required numbers $=10 \times 20=200$