The seven letters of the word NUMBERS are written on seven cards, and the cards are placed in a row…

Step 1. All arrangements. $7! = 5040$.

Step 2. Vowels together. The block and the five consonants are $6$ objects, which can be arranged in $6! = 720$ ways. The vowels swap inside the block in $2$ ways. So $720 \times 2 = 1440$.

Step 3. The probability. $\dfrac{1440}{5040} = \dfrac{2}{7} \approx 0.2857$.

Step 4. Consonant first, vowel last. The last place: $2$ vowels. The first place: $5$ consonants. The middle five places: $5! = 120$ ways. So $2 \times 5 \times 120 = 1200$ arrangements, and $\dfrac{1200}{5040} = \dfrac{5}{21} \approx 0.2381$.