$INVOLUTE$ शब्द के अक्षरों से, अर्थपूर्ण या अर्थहीन प्रत्येक $3$ स्वरों तथा $2$ व्यंजनों वाले, कितने शब्दों की रचना की जा सकती है ?

In the word $INVOLUTE$, there are $4$ vowels, namely, $I,O,E,U$ and $4$ consonants, namely, $N , V , L$ and $T.$

The number of ways of selecting $3$ vowels out of $4=\,^{4} C _{3}=4$

The number of ways of selecting $2$ consonants out of $4=\,^{4} C _{2}=6$

Therefore, the number of combinations of $3$ vowels and $2$ consonants is $4 \times 6=24$

Now, each of these $24$ combinations has $5$ letters which can be arranged among themselves in $5 !$ ways. Therefore, the required number of different words is $24 \times 5 !=2880$