The number of 4-letter words, with or without | vedclass

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(B) The word '$PQRPQRSTUVP$' contains $11$ letters: $P(3), Q(3), R(2), S(1), T(1), U(1), V(1)$.Wait, let us re-count: $P, Q, R, P, Q, R, S, T, U, V, P

Vedclass · Accepted Answer

$1422$

Vedclass · Answer

(B) The word '$PQRPQRSTUVP$' contains $11$ letters: $P(3), Q(3), R(2), S(1), T(1), U(1), V(1)$.Wait, let us re-count: $P, Q, R, P, Q, R, S, T, U, V, P$. Letters are: $P: 3, Q: 2, R: 2, S: 1, T: 1, U: 1, V: 1$. Total distinct letters are $7$ $(P, Q, R, S, T, U, V)$.Case $I$: $3$ alike, $1$ different: Choose $1$ letter from ${P}$ to be $3$ alike, and $1$ from remaining $6$ letters. Number of ways $= {}^{1}C_{1} 	imes {}^{6}C_{1} 	imes \frac{4!}{3!} = 1 	imes 6 	imes 4 = 24$.Case $II$: $2$ alike, $2$ alike: Choose $2$ letters from ${P, Q, R}$ to be $2$ alike each. Number of ways $= {}^{3}C_{2} 	imes \frac{4!}{2!2!} = 3 	imes 6 = 18$.Case $III$: $2$ alike, $2$ different: Choose $1$ letter from ${P, Q, R}$ to be $2$ alike, and $2$ from remaining $6$ letters. Number of ways $= {}^{3}C_{1} 	imes {}^{6}C_{2} 	imes \frac{4!}{2!} = 3 	imes 15 	imes 12 = 540$.Case $IV$: All $4$ different: Choose $4$ letters from $7$ distinct letters. Number of ways $= {}^{7}C_{4} 	imes 4! = 35 	imes 24 = 840$.Total words $= 24 + 18 + 540 + 840 = 1422$.

The number of $4$-letter words, with or without meaning, which can be formed using the letters of '$PQRPQRSTUVP$',is:

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