Using tree diagrams to enumerate parallelable occurrence of an event

Tree Diagrams
To register all members of an event, sometimes we can use a tool like diagram which is called tree diagram, as on the following example.

Example 2:
Determine the number of numbers which contain of three digit and its value is less than 300, if these number is formed with the numeral: 2, 3, 4, 5, 6, and the numeral used can’t be repeated.

Because the numbers which will be determined <300, so to form the hundreds from the numeral the available is 2. Furthermore, to form the tens and units are free as long as there is no repetation of numeral. The numbers that can be formed, can be drawn with the following tree diagram.

using tree diagrams to generate license plates alphanumeric
using tree diagrams to generate license plates alphanumeric

The numbers are:
234, 235, 236, 243, 245, 246, 253, 254, 256, 263, 264, and 265
the number of numbers is 12.

Exercise Competency Test 2

1. Numbers is consist of three different numeral, and their value less than 400 and they are formed with numeral: 3, 4, 5, 6. the number of numbers that formed is ….
A. 2 B. 3 C. 5 D. 6 E. 9
Answer: D, explanation:
The numbers start from the available numeral is 3.
The numbers are:
345, 346, 354, 356, 364, 365
the number of numbers is 6.

2. Numbers is consist of three different numeral, and they are even number, and they are formed with numeral: 2, 3, 4, 5, 6. The number of numbers that formed is ….
A. 3 B. 5 C. 6 D. 10 E. 36
Answer: explanation:
even numbers are finished by {0, 2, 4, 6, 8}, but in this section they are finished by the available numeral are {2, 4, 6}. Thus forming of numbers need to start from even numeral (numeral of the most right).

…←⋯←{2,4,6}
 

enumerating even-numbers
enumerating even-numbers

The numbers are:

234, 236, 246, 254, 256, 264,
324, 326, 342, 346, 352, 354,
356, 362, 364, 426, 432, 436,
452, 456, 462, 524, 526, 532,
534, 536, 542, 546, 562, 564,
624, 632, 634, 642, 652, 654

What! The number of numbers is 36.

3. A coin is tossed four times. The number of possible ways of the appears of coin’s side is ….
A. 4 B. 8 C. 10 D. 16 E. 32
Answer: D, explanation:
The coin has two sides, those are a number and an image. Suppose N denote the appearing is number, and E denote the appearing is image. The possible ways for tossing once are:

{E,N}→2→21

The possible ways for tossing twice are:

{EE,EN,NE,NN}→4→22

The number of possible ways for tossing n times is 2n. Thus, the number of possible ways for tossing 4 times is 24=16.

4. A student has three shirts and two pants. The shirt and pant can be used to the school. The number of way that student to make pairs the shirt and pant is ….
A. 2 B. 4 C. 5 D. 6 E. 8
Answer: D, explanation:
Suppose S denote the shirt then {S1,S2,S3} denote each of the shirts. Suppose P denote the pant then {P1,P2} denote each of the pants. The possible pairs are:

{S1P1,S1P2,S2P1,S2P2,S3P1,S3P2}

The number of way that student to make pairs the shirt and pant is nSnP=3∙2=6

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