Question

(1) (a) How many distinct 5-digit numbers (i.e. numbers between 20000 and 50000) cul be forni(XI using lhe digits { I, %, 3, 4, 5, 6 such lhat no digils repcai'. (Hint: Count all the distinct numbers with no restriction on the leading (most significant) digit and then subtract the count of numbers with restrictions on the 1st digit) (b) (i) How many arrangements are there of the letters of MATCH (i) Of these, how many have the letters M and A adjacent? (i) How many have the letters M and A side-by-side in the order AM

Answer 1

1^st 2^nd 3^rd 4^th 5^th There are 6 choices for 1^st place v[z] 1 to 6 There a_x 5 choices for 2^nd place There a_x 4 choices for 3^rd place There a_x 3 choices for 4^th place There a_x 2 choices for 5^th place therefore By multiplication principle Total no. of digits with no restriction = 6 middot 5 middot 4 middot 3 middot 2 = 720 But, 5 digit no. should be in between 200000 to 500000 therefore 1^st place has 2, 3, 4, 5 i.e., 4 choices & remaining places choices same as above. Therefore Total numbers with restriction on 1^st digit = 4 middot 5 middot 4 middot 3 middot 2 = 480 therefore Required total no.s = 720 - 480 = 240 \r\n There are 5 letters in the word MATCH & all letters are distinct. Therefore Total arrangements = 5! = 120 Consider a group of A & M. \r\n < null >