Question

(Tags) One planet has 3¹⁰ inhabitants, each of which is identified by 10 characters-tag. Each character in the tags is either A, B, or C (with different tags)
and each resident has a different identifier (so all tags are
currently in use). Two examples of tags: CBABBABBAC, AABABCAACC.
(a) How many residents have a palindrom tag? (i.e., the same as the front and back,
e.g. AABCAACBAA)
(b) How many inhabitants of the label do not have the same two consecutive characters?
(eg ABACABACAB is like this, while AACBCBABCB is not)
Selecting a random resident of the planet (the chosen resident is therefore equally likely
any of the planet's inhabitants). With what probability,
(c) his identification is palindrom?
(d) the two consecutive characters of his identifier are not the same

Answer 1

the following two images have the complete solution with all reasoning and explanations.So check it out

image 1:

image 2:

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