Question

1.

A) How many three-digit numbers are there for which the sum of the digits is at least 25?

B) How many three-digit numbers can be formed if only odd numbers are allowed to be re-used

Please combinatorics principles where applicable.

Answer 1

Number should be 3-digit number Sumo了 eligits should be a+ teas t 2s 3725 aximum poss ible 3-digit n.m ben. İS, uhan an 3 digits are 9 maximum sum2 Possible Cases (ay when sum is 25 These 3 digits car be awangea ameny them selves in 3! uays 2 2l Those 3 numbeys ax Thus 3 numbos ib) 9十9十3と 25 can be aman ged in 31 wayS Those numbeYS are, 3 9 299 C) Sum is 26 only one possibility 9,srepeated 3 Total possible numbers 2 Those numbers are 3

d) sum is 23 only one possibility is thee nvmbey er possible nombm3 τ 3-digit num bes sith sum 2 25 i hus, total num bey 3+3+3+1 B) only odd numbers can be reused case (a) is elimin뎌忙d , as theve is only one ockl nun, bev and even number 15 xe peated. case (b).odd numbes q is reused, so total 3 possibiliHes. Case (c)odd number is reused so total 3 possibilibes cose (d) eda rumbev 9 í, า?used and on,di possibihty Theve fore, total 3chgit num bess where on odd numbers oct cl avert peated 3+3+ τ digit num beys.