Combinatorics:01

প্রশ্ন:1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 একক দৈর্ঘ্য বিশিষ্ট রেখা সমূহ দ্বারা কয়টি ত্রিভুজ গঠন সম্ভব???

Solution:

দেওয়া আছে,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
আমরা জানি,ত্রিভুজ গঠনের জন্য,দুইটি বাহুর যোগফল,তৃতীয় টির চেয়ে বেশি হতে হবে। অর্থাৎ,a+b>c

16 টি হতে যেকোনো তিনটি নিয়ে ত্রিভুজ গঠন সম্ভব=16C3।এখন আমরা বের করবো,কোনগুলো তে ত্রিভুজ গঠন অসম্ভব

Case: a+b=c or a+b<c হলে ত্রিভুজ গঠন অসম্ভব। (I)বাহু a=1,b=2 হলে,c এর মান সম্ভব 14 টি
a=1,b=3 হলে c এর মান সম্ভব=13 টি...... .....
..... a=1,b=15 হলে,c সম্ভব=1 টি
অর্থাৎ,a=1 fixed রেখে মোট অসম্ভব ত্রিভুজ সংখ্যা=1+2+3+....+14=15C2[we know→1+2+....+n=(n+1)C2]

a=2 fixed রেখে মোট অসম্ভব ত্রিভুজ সংখ্যা=1+2+3+....+12=13C2
a=3 fixed রেখে,1+2+...10=11C2
এভাবে a=7 fixed করে পর্যন্ত চলতে থাকবে।ফলে,

মোট অসম্ভব ত্রিভুজ সংখ্যা=15C2+13C2+11C2+.....+3C2
Ans:16C3-(15C2+13C2+11C2+....+3C2)
=252


Translation:

Q: How many scalene triangles with sides a,b,ca, b , c can be formed, where a,b,c{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}a, b, c \in \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\}?

Solution:

We know that, for the formation of the triangle, the sum of two sides, will be more than a third. In other words, a + b> c

Total ways of choosing three numbers = 16 C3. Now we'll find combinations which are impossible to form a triangle.

If a+b=ca + b = c or a+b<ca + b <c then it is impossible to form a triangle.
If a = 1, b = 2, then, there are 14 possible values of cc.
If a = 1, b = 3, then there are 13 possible values of cc .
\vdots \quad \quad \vdots \quad \quad \vdots
If a = 1, b = 15, then, 1 value of cc is possible.
In other words, when a = 1 (fixed), the total number of impossible triangles = 1 + 2 + 3 + .... + 14 = 15 C2 [we know → 1 + 2 + .... + n = (n + 1) C2]

With a = 2 fixed, the total number of impossible triangles = 1 + 2 + 3 + .... + 12 = 13 C2
With a = 3 fixed, 1 + 2 + ... 10 = 11 C2
Thus, if a = 7 fixed, 1 + 2 = 3 C2

The total number of impossible triangles = 15 C2 + 13C2 + 11C2 + ..... + 3C2
Ans: The total number of possible triangles = 16C3- (15C2 + 13C2 + 11C2 + .... + 3C2)
= 252

Note by Sazzad Hossain Rafi
3 years, 3 months ago

No vote yet
1 vote

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