The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is ________ .

Suppose that a function f : R $$ \to $$ R satisfies
f(x + y) = f(x)f(y) for all x, y $$ \in $$ R and f(1) = 3.
If $$\sum\limits_{i = 1}^n {f(i)} = 363$$ then n is equal to ________ .

If $$\overrightarrow x $$ and $$\overrightarrow y $$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ and $${2\overrightarrow x + \lambda \overrightarrow y }$$ is perpendicular to $${\overrightarrow y }$$, then the value of $$\lambda $$ is _________ .

Consider the data on x taking the values
0, 2, 4, 8,....., 2ⁿ with frequencies
ⁿC₀ , ⁿC₁ , ⁿC₂ ,...., ⁿC_n respectively. If the
mean of this data is $${{728} \over {{2^n}}}$$, then n is equal to _________ .