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Let $N=\{1,2, \ldots, 9\}$ and $L=\{a, b, c\}$ which if the following is correct ?

  • $L \cup N$ is arranged on a line with the letters appearing consecutively (in any order). The number of such arrangements are less than $10 ! \times 5$.
  • More than half of the functions from $N$ to $L$ have $b$ in their range.
  • The number of one-to-one functions from $L$ to $N$ is less than 512 .
  • The number of functions $N$ to $L$ that do not map consecutive numbers to consecutive letters is greater than $512 .$

i have checked the first three parts as such for first one its false since it should be haviing 3! = 6 instead of 5 there for total permutations of L elements . Next one is simply $3^9 - 2^9$ by complementary couting . Third one is the $\binom{9}{3} 3!$ (less than 512) , next the final option i am not getting the statement at all , i think(for complementary counting) it means we must have all terms having consecutive letters as output but there are only 3 letters so how can it be possible ? Or is that means answer is same as total number of functions itself ?

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  • $\begingroup$ A compliment is an expression of praise. The complement of a set $A$ in the universe $U$ is the set $U \setminus A$ consisting of elements in the universe that are not in $A$. You meant complementary. The way I read the last question is that if $f(k) = a$, then $f(k + 1) \neq b$ and that if $f(k) = b$, then $f(k + 1) \neq c$. $\endgroup$ May 28 at 10:07
  • $\begingroup$ I see indeed i rectified and yeah now makes sense , but how will we solve for all possible type of that one ? I would do like case by case of having either 1 to be a etc.. ? $\endgroup$
    – Orion_Pax
    May 28 at 13:02
  • $\begingroup$ I think i got a method for getting a lower bound on it , please for once check my solution here @N.F.Taussig which i am sharing here soon $\endgroup$
    – Orion_Pax
    May 29 at 5:54

1 Answer 1

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Let $W_n$ denotes the number of functions having first $'n'$ natural numbers as domain and co domain as ${a,b,c}$ having first element mapping to $a$ , similarily $Z_n$ denotes mapping in which first element is map to $b$ . And $Q_n$ denotes functions in which the first element mapping is with $c$ , then as we know that if anywhere the element mapping is a then next element mapping will either be $a$ or $c$ but not $b$ , similarily for other two $b$ and $c$ ones we get the recurrence relations of
$Z_n = Z_{n-1} + W_{n-1}$ ,$Q_n = Q_{n-1} + Z_{n-1} + W_{n-1} $ and $W_n = Q_{n-1} + W_{n-1}$ .

Total functions $\phi(n)$ for a specific $n$ would be of form = $2\phi(n-1) + W_{n-1}$ , as can observe the $W_{n-1}$ cannot be negative therefore lower bound is $2\phi(n-1)$ , as $\phi(1) = 3$ , therefore $\phi(9)$ is atleast $3×2^8$ which is already greater than $2^9$ since $3>2$

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  • $\begingroup$ You should explain why the answer is at least $3 \cdot 2^8$. In particular, you should explain that a $c$ can be followed by any of the three letters. $\endgroup$ May 29 at 9:41

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