Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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.

Permutations and Combinations

Solution:

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 $=10 \times 20=200$