Using the digits $1,2,3,4,5,6,7$, a number of 4 different digits is formed. Find
| Column I | Column II | ||
|---|---|---|---|
| (i) | how many numbers are formed? | (a) | 840 |
| (ii) | how many numbers are exactly divisible by 2? | (b) | 200 |
| (iii) | how many numbers are exactly divisible by 25? | (c) | 360 |
| (iv) | how many of these are exactly divisible by 4? | (d) | 40 |
(i) Total numbers of 4 digit formed with digits 1, 2, 3, 4, 5, 6, 7
$$=7 \times 6 \times 5 \times 4=840$$
(ii) When a number is divisible by 2 . At its unit place only even numbers occurs.
$$\text { Total numbers }=4 \times 5 \times 6 \times 3=360$$
(iii) Total numbers which are divisible by $25=40$
(iv) A number is divisible by 4 , If its last two digit is divisible by 4 .
$\therefore$ Total such numbers $=200$
How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if
| Column I | Column II | ||
|---|---|---|---|
| (i) | 4 letters are used at a time | (a) | 720 |
| (ii) | All letters are used at a time | (b) | 240 |
| (iii) | All letters are used but the first is a vowel. | (c) | 360 |
(i) 4 letters are used at a time $={ }^6 P_4=\frac{6!}{2!}=6 \times 5 \times 4 \times =360$
(ii) All letters used at a time $=6$ ! $=6 \times 5 \times 4 \times 3 \times 2 \times 1=720$
(iii) All letters used but first is vowel $=2 \times 5$ ! $=2 \times 5 \times 4 \times 3 \times 2 \times 1=240$