The number of distinct five-letter arrangements (words, meaningful or not) that can be formed from the letters of the word ENGINEER, in which both…

ENGINEER has letters E (3), N (2), G, I, R. The vowels are E (3 copies) and I (1 copy). Casework on the end letters. (i) Both ends E: the middle three letters come from the remaining E (1), N (2), G, I, R. Count 3-letter arrangements from {E, N, N, G, I, R}: all distinct letters from 5 kinds give $5\cdot4\cdot3=60$; arrangements using N twice: choose the third letter from 4 others and place it in 3 ways, giving $12$; total $72$. (ii) One end E and the other end I: 2 orders; middle uses {E,E,N,N,G,R}: distinct from 4 kinds gives $24$, one repeated letter (E or N) gives $2\cdot3\cdot3=18$, total $42$, so $2\cdot42=84$. Overall $72+84=156$, matching an exhaustive enumeration.