In a certain laboratory, chemicals are identified by a colour-coding system. There are 20 different chemicals. Each one is coded with either a single colour or a unique two-colour pair. If the order of colours in the pairs does not matter, what is the minimum number of different colours needed to code all 20 chemicals with either a single colour or a unique pair of colours ?
Answer: B Each one coded with either a single colour or unique two-colour pair. Therefore, total number of ways = n + nC2 Minimum number of different colour needed to code all 20 chemicals will be 6. => 6 +6C2 = 21.
Q. No. 32:
How many positive integers 'n' can be form using the digits 3,4,4,5,6,6,7, if we want 'n' to exceed 60,00,000 ?
Answer: C As per the given condition, number in the highest position should be either 6 or 7, which can be done in 2 ways. If the first digit is 6, the other digits can be arranged in 6!/2! = 360 ways. If the first digit is 7, the other digits can be arranged in 6!/(2!*2!) = 180 ways. Thus required possibilities for n = 360+180 = 540 ways.
Q. No. 33:
In how many ways can four letters of the word 'SERIES' be arranged ?
Answer: D The given word 'SERIES' contains 2S, 2E and rest are distinct. The number of ways of selecting the 4 letter and the number of arrangement are as follows : Case I : 4 letters are distinct = S,E,R,I = 4! = 24 ways. Case II : 2 letter are same and 2 letter are distinct = {SSRI, SSRE, SSIE, EERI, EERS and EEIS} = 4!/2! * 6 = 72 ways. Case III : Two are same of one kind and two are same of the other kind = SSEE = 4!/(2!*2!) = 6 ways. Total number of ways = 24 + 72 + 6 = 102.
Q. No. 34:
While packing for a business trip Mr. Debashis has packed 3 pairs of shoes, 4 pants, 3 half-pants, 6 shirts, 3 sweater and 2 jackets. The outfit is defined as consisting of a pair of shoes, a choice of "lower wear" (either a pair or a half-pant), a choice of "upper wear" (it could be a shirt or sweater or both) and finally he may or may not choose to wear a jacket. How many different outfits are possible ?
Answer: D Shoes can be selected = 3 ways. Lower wear can be selected = (4+3) = 7 ways. Upper wear can be selected = (6 + 3+ 9*2) =27 ways. Jacket can be selected = 2 ways. He may not wear jacket = 3ways. Total possible outputs = 3*7*27*3 = 1701 ways.